| L(s) = 1 | + 64·2-s + 486·3-s + 3.07e3·4-s − 2.41e3·5-s + 3.11e4·6-s − 7.35e4·7-s + 1.31e5·8-s + 1.77e5·9-s − 1.54e5·10-s − 5.34e5·11-s + 1.49e6·12-s + 7.42e5·13-s − 4.70e6·14-s − 1.17e6·15-s + 5.24e6·16-s − 7.30e6·17-s + 1.13e7·18-s + 4.43e5·19-s − 7.40e6·20-s − 3.57e7·21-s − 3.41e7·22-s − 5.17e7·23-s + 6.37e7·24-s − 9.03e7·25-s + 4.75e7·26-s + 5.73e7·27-s − 2.25e8·28-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 1.15·3-s + 3/2·4-s − 0.344·5-s + 1.63·6-s − 1.65·7-s + 1.41·8-s + 9-s − 0.487·10-s − 1.00·11-s + 1.73·12-s + 0.554·13-s − 2.33·14-s − 0.398·15-s + 5/4·16-s − 1.24·17-s + 1.41·18-s + 0.0410·19-s − 0.517·20-s − 1.91·21-s − 1.41·22-s − 1.67·23-s + 1.63·24-s − 1.85·25-s + 0.784·26-s + 0.769·27-s − 2.48·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6084 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6084 ^{s/2} \, \Gamma_{\C}(s+11/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_1$ | \( ( 1 - p^{5} T )^{2} \) |
| 3 | $C_1$ | \( ( 1 - p^{5} T )^{2} \) |
| 13 | $C_1$ | \( ( 1 - p^{5} T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 + 482 p T + 3846106 p^{2} T^{2} + 482 p^{12} T^{3} + p^{22} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 73554 T + 629158402 p T^{2} + 73554 p^{11} T^{3} + p^{22} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 534316 T + 534217739782 T^{2} + 534316 p^{11} T^{3} + p^{22} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 7307908 T + 46255899355606 T^{2} + 7307908 p^{11} T^{3} + p^{22} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 443466 T + 121756190247502 T^{2} - 443466 p^{11} T^{3} + p^{22} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 2247912 p T + 1796149027827598 T^{2} + 2247912 p^{12} T^{3} + p^{22} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 82966440 T + 13977623951807158 T^{2} + 82966440 p^{11} T^{3} + p^{22} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 234967130 T + 63779165097985326 T^{2} + 234967130 p^{11} T^{3} + p^{22} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 180419688 T + 85653224220175558 T^{2} + 180419688 p^{11} T^{3} + p^{22} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 86491574 T - 175455811617756278 T^{2} - 86491574 p^{11} T^{3} + p^{22} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 829328952 T + 1495257681670922614 T^{2} + 829328952 p^{11} T^{3} + p^{22} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 1601234704 T + 5371784024377066846 T^{2} + 1601234704 p^{11} T^{3} + p^{22} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 153096044 T - 8322908555225502722 T^{2} - 153096044 p^{11} T^{3} + p^{22} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 2054293064 T - 25552466176148821802 T^{2} + 2054293064 p^{11} T^{3} + p^{22} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 9856427904 T + \)\(11\!\cdots\!62\)\( T^{2} - 9856427904 p^{11} T^{3} + p^{22} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 14603793338 T + \)\(12\!\cdots\!02\)\( T^{2} + 14603793338 p^{11} T^{3} + p^{22} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 34487313860 T + \)\(74\!\cdots\!42\)\( T^{2} + 34487313860 p^{11} T^{3} + p^{22} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 6870504440 T + \)\(32\!\cdots\!18\)\( T^{2} + 6870504440 p^{11} T^{3} + p^{22} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 27071619800 T + \)\(16\!\cdots\!94\)\( T^{2} + 27071619800 p^{11} T^{3} + p^{22} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 57969138432 T + \)\(32\!\cdots\!46\)\( T^{2} - 57969138432 p^{11} T^{3} + p^{22} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 68240894690 T + \)\(66\!\cdots\!22\)\( T^{2} - 68240894690 p^{11} T^{3} + p^{22} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 168488621496 T + \)\(21\!\cdots\!94\)\( T^{2} + 168488621496 p^{11} T^{3} + p^{22} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.99069558140997090747130670615, −11.70049069353305475844381150037, −10.73186268784086229940211846744, −10.40134546812842036241113878987, −9.534272806611928378693034395237, −9.359218575222549799906739153420, −8.151967895742476093950598201707, −8.036083462649139555963073801575, −7.06630331733824805730654151883, −6.75226627193807188734794387501, −5.86580844786814835477997649354, −5.57456403336268758144234240569, −4.27458182057639836764907176615, −4.09225218059209269800440246993, −3.21389933288216272487006535555, −3.13193835447252211046096640788, −1.93512502682970730678956776149, −1.93000801477335972721182138738, 0, 0,
1.93000801477335972721182138738, 1.93512502682970730678956776149, 3.13193835447252211046096640788, 3.21389933288216272487006535555, 4.09225218059209269800440246993, 4.27458182057639836764907176615, 5.57456403336268758144234240569, 5.86580844786814835477997649354, 6.75226627193807188734794387501, 7.06630331733824805730654151883, 8.036083462649139555963073801575, 8.151967895742476093950598201707, 9.359218575222549799906739153420, 9.534272806611928378693034395237, 10.40134546812842036241113878987, 10.73186268784086229940211846744, 11.70049069353305475844381150037, 11.99069558140997090747130670615
