L(s) = 1 | + 2-s + 1.41·3-s + 4-s + 1.41·6-s + 8-s − 0.999·9-s − 2·11-s + 1.41·12-s + 16-s − 7.07·17-s − 0.999·18-s + 1.41·19-s − 2·22-s + 1.41·24-s − 5·25-s − 5.65·27-s − 6·29-s + 32-s − 2.82·33-s − 7.07·34-s − 0.999·36-s − 2·37-s + 1.41·38-s − 7.07·41-s − 6·43-s − 2·44-s − 2.82·47-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.816·3-s + 0.5·4-s + 0.577·6-s + 0.353·8-s − 0.333·9-s − 0.603·11-s + 0.408·12-s + 0.250·16-s − 1.71·17-s − 0.235·18-s + 0.324·19-s − 0.426·22-s + 0.288·24-s − 25-s − 1.08·27-s − 1.11·29-s + 0.176·32-s − 0.492·33-s − 1.21·34-s − 0.166·36-s − 0.328·37-s + 0.229·38-s − 1.10·41-s − 0.914·43-s − 0.301·44-s − 0.412·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5194 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5194 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| 53 | \( 1 - T \) |
good | 3 | \( 1 - 1.41T + 3T^{2} \) |
| 5 | \( 1 + 5T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + 7.07T + 17T^{2} \) |
| 19 | \( 1 - 1.41T + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 + 7.07T + 41T^{2} \) |
| 43 | \( 1 + 6T + 43T^{2} \) |
| 47 | \( 1 + 2.82T + 47T^{2} \) |
| 59 | \( 1 + 1.41T + 59T^{2} \) |
| 61 | \( 1 - 11.3T + 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 - 12.7T + 73T^{2} \) |
| 79 | \( 1 + 12T + 79T^{2} \) |
| 83 | \( 1 + 9.89T + 83T^{2} \) |
| 89 | \( 1 - 9.89T + 89T^{2} \) |
| 97 | \( 1 - 4.24T + 97T^{2} \) |
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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
−7.907391988025840112195600170615, −7.10031646213773678597591475408, −6.42268292566425518859325936842, −5.53806868170035424956252457516, −4.93720037988371086196588641170, −3.94663319521752276784553730279, −3.37625594198578000932538242992, −2.41469691043021875125231859681, −1.89710798323623730877736978384, 0,
1.89710798323623730877736978384, 2.41469691043021875125231859681, 3.37625594198578000932538242992, 3.94663319521752276784553730279, 4.93720037988371086196588641170, 5.53806868170035424956252457516, 6.42268292566425518859325936842, 7.10031646213773678597591475408, 7.907391988025840112195600170615