| L(s) = 1 | + 3.26·3-s − 5·5-s − 27.7·7-s − 16.3·9-s + 10.8·11-s + 36.9·13-s − 16.3·15-s + 118.·17-s + 19.3·19-s − 90.6·21-s − 23·23-s + 25·25-s − 141.·27-s + 234.·29-s − 165.·31-s + 35.2·33-s + 138.·35-s + 202.·37-s + 120.·39-s − 295.·41-s + 65.9·43-s + 81.8·45-s + 110.·47-s + 429.·49-s + 387.·51-s − 688.·53-s − 54.0·55-s + ⋯ |
| L(s) = 1 | + 0.627·3-s − 0.447·5-s − 1.50·7-s − 0.606·9-s + 0.296·11-s + 0.788·13-s − 0.280·15-s + 1.69·17-s + 0.233·19-s − 0.941·21-s − 0.208·23-s + 0.200·25-s − 1.00·27-s + 1.50·29-s − 0.957·31-s + 0.186·33-s + 0.671·35-s + 0.898·37-s + 0.494·39-s − 1.12·41-s + 0.233·43-s + 0.271·45-s + 0.342·47-s + 1.25·49-s + 1.06·51-s − 1.78·53-s − 0.132·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + 5T \) |
| 23 | \( 1 + 23T \) |
| good | 3 | \( 1 - 3.26T + 27T^{2} \) |
| 7 | \( 1 + 27.7T + 343T^{2} \) |
| 11 | \( 1 - 10.8T + 1.33e3T^{2} \) |
| 13 | \( 1 - 36.9T + 2.19e3T^{2} \) |
| 17 | \( 1 - 118.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 19.3T + 6.85e3T^{2} \) |
| 29 | \( 1 - 234.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 165.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 202.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 295.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 65.9T + 7.95e4T^{2} \) |
| 47 | \( 1 - 110.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 688.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 10.5T + 2.05e5T^{2} \) |
| 61 | \( 1 - 110.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 643.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 143.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 158.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.12e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 824.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 879.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 938.T + 9.12e5T^{2} \) |
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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
−8.436451256620267781899990791460, −7.895084759745822874599303021102, −6.89706127053934023141818461964, −6.15103283406791307081019789240, −5.40561001661631117013055518849, −3.98842550963524995683065528868, −3.31559684389415897181064917731, −2.80273094126920966076928081073, −1.20889117811910789382085883980, 0,
1.20889117811910789382085883980, 2.80273094126920966076928081073, 3.31559684389415897181064917731, 3.98842550963524995683065528868, 5.40561001661631117013055518849, 6.15103283406791307081019789240, 6.89706127053934023141818461964, 7.895084759745822874599303021102, 8.436451256620267781899990791460
