| L(s) = 1 | − 2·2-s + 3·4-s − 4·8-s − 6·9-s − 8·11-s + 5·16-s + 12·18-s + 16·22-s − 8·29-s − 6·32-s − 18·36-s + 12·37-s + 24·43-s − 24·44-s + 24·53-s + 16·58-s + 7·64-s + 24·67-s + 16·71-s + 24·72-s − 24·74-s + 27·81-s − 48·86-s + 32·88-s + 48·99-s − 48·106-s − 24·107-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s − 1.41·8-s − 2·9-s − 2.41·11-s + 5/4·16-s + 2.82·18-s + 3.41·22-s − 1.48·29-s − 1.06·32-s − 3·36-s + 1.97·37-s + 3.65·43-s − 3.61·44-s + 3.29·53-s + 2.10·58-s + 7/8·64-s + 2.93·67-s + 1.89·71-s + 2.82·72-s − 2.78·74-s + 3·81-s − 5.17·86-s + 3.41·88-s + 4.82·99-s − 4.66·106-s − 2.32·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6002500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6002500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7493515527\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7493515527\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 80 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 72 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 128 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 122 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 160 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 176 T^{2} + p^{2} T^{4} \) |
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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
−9.017962518218447622973711357742, −8.901222597456340537824936197451, −8.155630361501301973507933802985, −8.072187216788958746792343613527, −7.71307990388623155206703160837, −7.65913089315641033612952037776, −6.76822526672299875172440083481, −6.72966153601372802836379355112, −5.77557023760977352702314898595, −5.76941291043056006657978115710, −5.32840735865856745731057750930, −5.25033001024093999824911844789, −4.07789556565919454446923342208, −3.95298278422723375473933362025, −2.97507406132077813014808994290, −2.73662606160235482856814400419, −2.34590131526935004577305853862, −2.14025401432212350924364929516, −0.67837311627686396574912265316, −0.58509110284273866434600719133,
0.58509110284273866434600719133, 0.67837311627686396574912265316, 2.14025401432212350924364929516, 2.34590131526935004577305853862, 2.73662606160235482856814400419, 2.97507406132077813014808994290, 3.95298278422723375473933362025, 4.07789556565919454446923342208, 5.25033001024093999824911844789, 5.32840735865856745731057750930, 5.76941291043056006657978115710, 5.77557023760977352702314898595, 6.72966153601372802836379355112, 6.76822526672299875172440083481, 7.65913089315641033612952037776, 7.71307990388623155206703160837, 8.072187216788958746792343613527, 8.155630361501301973507933802985, 8.901222597456340537824936197451, 9.017962518218447622973711357742
