| L(s) = 1 | + 2-s + 4-s + 8-s − 3·9-s − 2·11-s + 16-s − 7·17-s − 3·18-s − 2·22-s + 3·23-s + 6·29-s − 7·31-s + 32-s − 7·34-s − 3·36-s − 4·37-s − 7·41-s − 8·43-s − 2·44-s + 3·46-s − 7·47-s + 4·53-s + 6·58-s − 14·59-s − 14·61-s − 7·62-s + 64-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.353·8-s − 9-s − 0.603·11-s + 1/4·16-s − 1.69·17-s − 0.707·18-s − 0.426·22-s + 0.625·23-s + 1.11·29-s − 1.25·31-s + 0.176·32-s − 1.20·34-s − 1/2·36-s − 0.657·37-s − 1.09·41-s − 1.21·43-s − 0.301·44-s + 0.442·46-s − 1.02·47-s + 0.549·53-s + 0.787·58-s − 1.82·59-s − 1.79·61-s − 0.889·62-s + 1/8·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 7 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 7 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 7 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 11 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 - 7 T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.569472348540349397233793689874, −7.77025897366962506474181031251, −6.76079918638171732221799087040, −6.28199686675875963082954387564, −5.18380963059850512231937039981, −4.79925241399182758704346995006, −3.58498646728436060162760380755, −2.81386949170859431374348881380, −1.87784006282304426223052050136, 0,
1.87784006282304426223052050136, 2.81386949170859431374348881380, 3.58498646728436060162760380755, 4.79925241399182758704346995006, 5.18380963059850512231937039981, 6.28199686675875963082954387564, 6.76079918638171732221799087040, 7.77025897366962506474181031251, 8.569472348540349397233793689874
