| L(s) = 1 | + 4·2-s + 6·3-s + 12·4-s − 10·5-s + 24·6-s + 32·8-s + 27·9-s − 40·10-s − 20·11-s + 72·12-s − 42·13-s − 60·15-s + 80·16-s − 76·17-s + 108·18-s − 90·19-s − 120·20-s − 80·22-s + 44·23-s + 192·24-s + 75·25-s − 168·26-s + 108·27-s − 160·29-s − 240·30-s − 62·31-s + 192·32-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 1.15·3-s + 3/2·4-s − 0.894·5-s + 1.63·6-s + 1.41·8-s + 9-s − 1.26·10-s − 0.548·11-s + 1.73·12-s − 0.896·13-s − 1.03·15-s + 5/4·16-s − 1.08·17-s + 1.41·18-s − 1.08·19-s − 1.34·20-s − 0.775·22-s + 0.398·23-s + 1.63·24-s + 3/5·25-s − 1.26·26-s + 0.769·27-s − 1.02·29-s − 1.46·30-s − 0.359·31-s + 1.06·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160900 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 3 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 5 | $C_1$ | \( ( 1 + p T )^{2} \) |
| 7 | | \( 1 \) |
| good | 11 | $D_{4}$ | \( 1 + 20 T + 1612 T^{2} + 20 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 42 T + 4789 T^{2} + 42 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 76 T + 5704 T^{2} + 76 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 90 T + 15559 T^{2} + 90 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 44 T + 19252 T^{2} - 44 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 160 T + 54028 T^{2} + 160 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 2 p T + 57599 T^{2} + 2 p^{4} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 358 T + 89141 T^{2} + 358 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 36 T + 2552 p T^{2} + 36 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 134 T + 129969 T^{2} + 134 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 684 T + 264994 T^{2} + 684 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 16 T + 182818 T^{2} - 16 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 552 T + 476584 T^{2} + 552 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 1312 T + 869394 T^{2} + 1312 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 194 T + 533609 T^{2} - 194 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 380 T - 111452 T^{2} - 380 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 198 T + 782269 T^{2} + 198 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 126 T - 277529 T^{2} - 126 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 864 T + 336184 T^{2} + 864 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 184 T + 284548 T^{2} - 184 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 568 T + 1864602 T^{2} + 568 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−8.792285470771940500642346618084, −8.513678453165362539452807580063, −7.892999211897369426322314509571, −7.82564358898168311012916835300, −7.18908085315637188285561416723, −7.04250505278549899481218059792, −6.39999361961186539909257023375, −6.27478913993626794568383058350, −5.24097919911480548530762371733, −5.15495940077311761215706075639, −4.58069839095985953457724162531, −4.29197966112112526146040698217, −3.74157792393920078817410338404, −3.40944870457445259475871919529, −2.80326114074161334526396936676, −2.59184555811204894857868438134, −1.71437557809031232888612030267, −1.64636295052117747224668630163, 0, 0,
1.64636295052117747224668630163, 1.71437557809031232888612030267, 2.59184555811204894857868438134, 2.80326114074161334526396936676, 3.40944870457445259475871919529, 3.74157792393920078817410338404, 4.29197966112112526146040698217, 4.58069839095985953457724162531, 5.15495940077311761215706075639, 5.24097919911480548530762371733, 6.27478913993626794568383058350, 6.39999361961186539909257023375, 7.04250505278549899481218059792, 7.18908085315637188285561416723, 7.82564358898168311012916835300, 7.892999211897369426322314509571, 8.513678453165362539452807580063, 8.792285470771940500642346618084
