| L(s) = 1 | + 2.56·2-s − 1.43·4-s − 5·5-s − 24.1·8-s − 12.8·10-s + 6.24·11-s + 56.3·13-s − 50.4·16-s − 24.6·17-s + 90.7·19-s + 7.19·20-s + 16·22-s − 69.8·23-s + 25·25-s + 144.·26-s − 228.·29-s + 67.8·31-s + 64.2·32-s − 63.0·34-s − 58.8·37-s + 232.·38-s + 120.·40-s − 19.2·41-s + 365.·43-s − 8.98·44-s − 178.·46-s + 195.·47-s + ⋯ |
| L(s) = 1 | + 0.905·2-s − 0.179·4-s − 0.447·5-s − 1.06·8-s − 0.405·10-s + 0.171·11-s + 1.20·13-s − 0.787·16-s − 0.350·17-s + 1.09·19-s + 0.0804·20-s + 0.155·22-s − 0.633·23-s + 0.200·25-s + 1.08·26-s − 1.46·29-s + 0.393·31-s + 0.354·32-s − 0.317·34-s − 0.261·37-s + 0.991·38-s + 0.477·40-s − 0.0731·41-s + 1.29·43-s − 0.0307·44-s − 0.573·46-s + 0.605·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + 5T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 2.56T + 8T^{2} \) |
| 11 | \( 1 - 6.24T + 1.33e3T^{2} \) |
| 13 | \( 1 - 56.3T + 2.19e3T^{2} \) |
| 17 | \( 1 + 24.6T + 4.91e3T^{2} \) |
| 19 | \( 1 - 90.7T + 6.85e3T^{2} \) |
| 23 | \( 1 + 69.8T + 1.21e4T^{2} \) |
| 29 | \( 1 + 228.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 67.8T + 2.97e4T^{2} \) |
| 37 | \( 1 + 58.8T + 5.06e4T^{2} \) |
| 41 | \( 1 + 19.2T + 6.89e4T^{2} \) |
| 43 | \( 1 - 365.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 195.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 511.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 284T + 2.05e5T^{2} \) |
| 61 | \( 1 - 123.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 144.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 73.0T + 3.57e5T^{2} \) |
| 73 | \( 1 + 638.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 976.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 484.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.01e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.80e3T + 9.12e5T^{2} \) |
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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
−8.313986404571865142652630448011, −7.52195641179090705481838590446, −6.52207133860112242905809928598, −5.79689757782884252790490275939, −5.11019922436989186607765222217, −4.03945007081128344518315577254, −3.69724288265100475036162569025, −2.66483754607630004393587062209, −1.23956327088675683827205019265, 0,
1.23956327088675683827205019265, 2.66483754607630004393587062209, 3.69724288265100475036162569025, 4.03945007081128344518315577254, 5.11019922436989186607765222217, 5.79689757782884252790490275939, 6.52207133860112242905809928598, 7.52195641179090705481838590446, 8.313986404571865142652630448011
