L(s) = 1 | − 2-s + 4-s + 5-s + 7-s − 8-s − 10-s + 2·13-s − 14-s + 16-s + 2·17-s − 4·19-s + 20-s + 23-s + 25-s − 2·26-s + 28-s + 10·29-s − 4·31-s − 32-s − 2·34-s + 35-s − 6·37-s + 4·38-s − 40-s − 2·41-s − 8·43-s − 46-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.377·7-s − 0.353·8-s − 0.316·10-s + 0.554·13-s − 0.267·14-s + 1/4·16-s + 0.485·17-s − 0.917·19-s + 0.223·20-s + 0.208·23-s + 1/5·25-s − 0.392·26-s + 0.188·28-s + 1.85·29-s − 0.718·31-s − 0.176·32-s − 0.342·34-s + 0.169·35-s − 0.986·37-s + 0.648·38-s − 0.158·40-s − 0.312·41-s − 1.21·43-s − 0.147·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 - T \) |
good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−16.51514929221883, −15.94689524392997, −15.25944727212153, −14.83935020622631, −14.15631560319180, −13.66061612174352, −13.00265346415374, −12.35365708632713, −11.82813699350571, −11.16842000005920, −10.56280518695980, −10.18289876933400, −9.547986917484276, −8.754420051291203, −8.436965112019674, −7.861225088702118, −6.943661310142417, −6.557118797842187, −5.843637536451403, −5.117695722400159, −4.433822204529039, −3.448177728786904, −2.784503671120706, −1.797022356218766, −1.266737150916821, 0,
1.266737150916821, 1.797022356218766, 2.784503671120706, 3.448177728786904, 4.433822204529039, 5.117695722400159, 5.843637536451403, 6.557118797842187, 6.943661310142417, 7.861225088702118, 8.436965112019674, 8.754420051291203, 9.547986917484276, 10.18289876933400, 10.56280518695980, 11.16842000005920, 11.82813699350571, 12.35365708632713, 13.00265346415374, 13.66061612174352, 14.15631560319180, 14.83935020622631, 15.25944727212153, 15.94689524392997, 16.51514929221883