| L(s) = 1 | − 2-s + 2·3-s − 2·4-s − 2·6-s + 5·7-s + 3·8-s − 4·11-s − 4·12-s + 13-s − 5·14-s + 16-s + 2·17-s − 3·19-s + 10·21-s + 4·22-s − 2·23-s + 6·24-s − 6·25-s − 26-s − 5·27-s − 10·28-s − 12·29-s − 7·31-s − 2·32-s − 8·33-s − 2·34-s − 7·37-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.15·3-s − 4-s − 0.816·6-s + 1.88·7-s + 1.06·8-s − 1.20·11-s − 1.15·12-s + 0.277·13-s − 1.33·14-s + 1/4·16-s + 0.485·17-s − 0.688·19-s + 2.18·21-s + 0.852·22-s − 0.417·23-s + 1.22·24-s − 6/5·25-s − 0.196·26-s − 0.962·27-s − 1.88·28-s − 2.22·29-s − 1.25·31-s − 0.353·32-s − 1.39·33-s − 0.342·34-s − 1.15·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 130299 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 130299 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - T + p T^{2} ) \) |
| 13 | $C_2$ | \( 1 - T + p T^{2} \) |
| 257 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 27 T + p T^{2} ) \) |
| good | 2 | $D_{4}$ | \( 1 + T + 3 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 4 T + 5 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 2 T + 15 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 23 | $D_{4}$ | \( 1 + 2 T + 15 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 12 T + 89 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 7 T + 50 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 7 T + 62 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $D_{4}$ | \( 1 + T - 30 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 14 T + 102 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $D_{4}$ | \( 1 - 10 T + 111 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 3 T + 32 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + T + 34 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 5 T + 52 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + T + 88 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 13 T + 154 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 146 T^{2} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 4 T + 37 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 13 T + 208 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
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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
−13.9450881442, −13.8129359604, −13.2122769763, −13.0191482636, −12.3606046136, −11.7265543205, −11.3316525816, −10.8219792024, −10.5304796117, −9.85118493680, −9.50942646394, −8.94700140863, −8.52091311163, −8.39614786187, −7.84000283004, −7.59823799076, −7.12904931739, −5.80150162738, −5.50513635690, −5.09015522881, −4.24059735226, −3.93366831843, −3.13330722974, −2.04911830345, −1.76284638152, 0,
1.76284638152, 2.04911830345, 3.13330722974, 3.93366831843, 4.24059735226, 5.09015522881, 5.50513635690, 5.80150162738, 7.12904931739, 7.59823799076, 7.84000283004, 8.39614786187, 8.52091311163, 8.94700140863, 9.50942646394, 9.85118493680, 10.5304796117, 10.8219792024, 11.3316525816, 11.7265543205, 12.3606046136, 13.0191482636, 13.2122769763, 13.8129359604, 13.9450881442
