| L(s) = 1 | + 2·2-s + 2·3-s + 3·4-s − 3·5-s + 4·6-s − 2·7-s + 4·8-s + 3·9-s − 6·10-s + 11-s + 6·12-s − 4·14-s − 6·15-s + 5·16-s − 10·17-s + 6·18-s + 3·19-s − 9·20-s − 4·21-s + 2·22-s − 10·23-s + 8·24-s + 25-s + 4·27-s − 6·28-s − 2·29-s − 12·30-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 1.15·3-s + 3/2·4-s − 1.34·5-s + 1.63·6-s − 0.755·7-s + 1.41·8-s + 9-s − 1.89·10-s + 0.301·11-s + 1.73·12-s − 1.06·14-s − 1.54·15-s + 5/4·16-s − 2.42·17-s + 1.41·18-s + 0.688·19-s − 2.01·20-s − 0.872·21-s + 0.426·22-s − 2.08·23-s + 1.63·24-s + 1/5·25-s + 0.769·27-s − 1.13·28-s − 0.371·29-s − 2.19·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50381604 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50381604 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 2 | $C_1$ | \( ( 1 - T )^{2} \) |
| 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| 13 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 + 3 T + 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - T + 18 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 19 | $D_{4}$ | \( 1 - 3 T + 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 10 T + 54 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 31 | $D_{4}$ | \( 1 - 2 T + 46 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - T - 32 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 65 T^{2} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 14 T + 118 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 3 T + 58 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 16 T + 153 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 18 T + 182 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 67 | $D_{4}$ | \( 1 + 6 T + 126 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 73 | $D_{4}$ | \( 1 + 9 T + 162 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 7 T + 166 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 4 T + 102 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 7 T + 152 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 8 T + 142 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
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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
−7.59799023034145298342820721618, −7.52418635389304660684300848863, −6.94035585208253992263829616745, −6.68729223779448540612542105646, −6.34675053330086894152983381453, −6.27606122046596693685449788097, −5.45507343282041274937272789542, −5.30594699752876541552305103223, −4.56163759182246706457856574962, −4.44480970558296690766779435603, −4.03425969554424684416716580851, −3.87875896133642508053166259367, −3.43373743379594914684767598537, −3.14553860213942786847966960567, −2.57719549039600103188700646321, −2.37217099840063851177879909996, −1.72644525597402823508401686277, −1.41161125840142754535699562107, 0, 0,
1.41161125840142754535699562107, 1.72644525597402823508401686277, 2.37217099840063851177879909996, 2.57719549039600103188700646321, 3.14553860213942786847966960567, 3.43373743379594914684767598537, 3.87875896133642508053166259367, 4.03425969554424684416716580851, 4.44480970558296690766779435603, 4.56163759182246706457856574962, 5.30594699752876541552305103223, 5.45507343282041274937272789542, 6.27606122046596693685449788097, 6.34675053330086894152983381453, 6.68729223779448540612542105646, 6.94035585208253992263829616745, 7.52418635389304660684300848863, 7.59799023034145298342820721618
