| L(s) = 1 | − 3-s + 9-s − 4·13-s − 17-s + 6·19-s − 3·23-s − 27-s + 5·31-s + 2·37-s + 4·39-s + 5·41-s + 2·43-s − 3·47-s + 51-s − 2·53-s − 6·57-s − 10·59-s − 12·67-s + 3·69-s + 3·71-s − 2·73-s − 9·79-s + 81-s − 8·83-s + 11·89-s − 5·93-s + 97-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s − 1.10·13-s − 0.242·17-s + 1.37·19-s − 0.625·23-s − 0.192·27-s + 0.898·31-s + 0.328·37-s + 0.640·39-s + 0.780·41-s + 0.304·43-s − 0.437·47-s + 0.140·51-s − 0.274·53-s − 0.794·57-s − 1.30·59-s − 1.46·67-s + 0.361·69-s + 0.356·71-s − 0.234·73-s − 1.01·79-s + 1/9·81-s − 0.878·83-s + 1.16·89-s − 0.518·93-s + 0.101·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 9 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 - 11 T + p T^{2} \) |
| 97 | \( 1 - T + p T^{2} \) |
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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
−15.52386253975076, −14.92882632530328, −14.30417713812759, −13.90527841139940, −13.28073293681652, −12.67145789910727, −12.15846399702773, −11.71633051934754, −11.29975802514534, −10.50609772675736, −10.10510709793043, −9.501039648475550, −9.116380520367829, −8.193641695421111, −7.624023059650573, −7.251520319807545, −6.495662029453379, −5.922886766682407, −5.355250409825919, −4.659509321328303, −4.277425629031518, −3.258983308772667, −2.690617968314383, −1.809996826158940, −0.9424819792400751, 0,
0.9424819792400751, 1.809996826158940, 2.690617968314383, 3.258983308772667, 4.277425629031518, 4.659509321328303, 5.355250409825919, 5.922886766682407, 6.495662029453379, 7.251520319807545, 7.624023059650573, 8.193641695421111, 9.116380520367829, 9.501039648475550, 10.10510709793043, 10.50609772675736, 11.29975802514534, 11.71633051934754, 12.15846399702773, 12.67145789910727, 13.28073293681652, 13.90527841139940, 14.30417713812759, 14.92882632530328, 15.52386253975076
