| L(s) = 1 | − 3·3-s − 19.1·5-s + 35.1·7-s + 9·9-s − 26·11-s + 13·13-s + 57.3·15-s − 36.2·17-s + 95.5·19-s − 105.·21-s − 161.·23-s + 240.·25-s − 27·27-s + 91.3·29-s − 266.·31-s + 78·33-s − 671.·35-s + 149.·37-s − 39·39-s − 77.8·41-s − 183.·43-s − 172.·45-s − 60.6·47-s + 890.·49-s + 108.·51-s − 281.·53-s + 496.·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.70·5-s + 1.89·7-s + 0.333·9-s − 0.712·11-s + 0.277·13-s + 0.987·15-s − 0.516·17-s + 1.15·19-s − 1.09·21-s − 1.46·23-s + 1.92·25-s − 0.192·27-s + 0.585·29-s − 1.54·31-s + 0.411·33-s − 3.24·35-s + 0.664·37-s − 0.160·39-s − 0.296·41-s − 0.651·43-s − 0.569·45-s − 0.188·47-s + 2.59·49-s + 0.298·51-s − 0.729·53-s + 1.21·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 3 | \( 1 + 3T \) |
| 13 | \( 1 - 13T \) |
| good | 5 | \( 1 + 19.1T + 125T^{2} \) |
| 7 | \( 1 - 35.1T + 343T^{2} \) |
| 11 | \( 1 + 26T + 1.33e3T^{2} \) |
| 17 | \( 1 + 36.2T + 4.91e3T^{2} \) |
| 19 | \( 1 - 95.5T + 6.85e3T^{2} \) |
| 23 | \( 1 + 161.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 91.3T + 2.43e4T^{2} \) |
| 31 | \( 1 + 266.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 149.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 77.8T + 6.89e4T^{2} \) |
| 43 | \( 1 + 183.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 60.6T + 1.03e5T^{2} \) |
| 53 | \( 1 + 281.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 542.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 65.0T + 2.26e5T^{2} \) |
| 67 | \( 1 - 1.03e3T + 3.00e5T^{2} \) |
| 71 | \( 1 - 1.04e3T + 3.57e5T^{2} \) |
| 73 | \( 1 - 483.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.33e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 812.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 936.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 954.T + 9.12e5T^{2} \) |
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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
−8.108760080133425991886987656202, −7.60017656197118777318539015898, −6.90494300873962263637923084598, −5.58102239378419334118463147405, −4.96771433169315622154168576267, −4.28186489066390762048437184519, −3.55647739449877737119131477014, −2.14411745501286284552865420270, −1.03265843656788454353178886262, 0,
1.03265843656788454353178886262, 2.14411745501286284552865420270, 3.55647739449877737119131477014, 4.28186489066390762048437184519, 4.96771433169315622154168576267, 5.58102239378419334118463147405, 6.90494300873962263637923084598, 7.60017656197118777318539015898, 8.108760080133425991886987656202
