| L(s) = 1 | − 2·2-s − 5.96·3-s + 4·4-s + 11.9·6-s + 15.2·7-s − 8·8-s + 8.62·9-s + 12.0·11-s − 23.8·12-s + 90.1·13-s − 30.4·14-s + 16·16-s + 83.9·17-s − 17.2·18-s + 74.8·19-s − 91.0·21-s − 24.0·22-s + 23·23-s + 47.7·24-s − 180.·26-s + 109.·27-s + 60.9·28-s + 94.5·29-s − 178.·31-s − 32·32-s − 71.7·33-s − 167.·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.14·3-s + 0.5·4-s + 0.812·6-s + 0.823·7-s − 0.353·8-s + 0.319·9-s + 0.329·11-s − 0.574·12-s + 1.92·13-s − 0.582·14-s + 0.250·16-s + 1.19·17-s − 0.225·18-s + 0.903·19-s − 0.945·21-s − 0.233·22-s + 0.208·23-s + 0.406·24-s − 1.36·26-s + 0.781·27-s + 0.411·28-s + 0.605·29-s − 1.03·31-s − 0.176·32-s − 0.378·33-s − 0.847·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.398310526\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.398310526\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2T \) |
| 5 | \( 1 \) |
| 23 | \( 1 - 23T \) |
| good | 3 | \( 1 + 5.96T + 27T^{2} \) |
| 7 | \( 1 - 15.2T + 343T^{2} \) |
| 11 | \( 1 - 12.0T + 1.33e3T^{2} \) |
| 13 | \( 1 - 90.1T + 2.19e3T^{2} \) |
| 17 | \( 1 - 83.9T + 4.91e3T^{2} \) |
| 19 | \( 1 - 74.8T + 6.85e3T^{2} \) |
| 29 | \( 1 - 94.5T + 2.43e4T^{2} \) |
| 31 | \( 1 + 178.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 296.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 193.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 15.4T + 7.95e4T^{2} \) |
| 47 | \( 1 - 128.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 342.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 725.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 784.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 782.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 141.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 404.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 721.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 985.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 463.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.68e3T + 9.12e5T^{2} \) |
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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
−9.427346487934332836565213412700, −8.575489513983043117314577891579, −7.85876996470315743675158079354, −6.93134510642987091352839699023, −5.88814764717059309422166354874, −5.57164932306202968725455985922, −4.29234122538525818419634575925, −3.11593373105241193282252090259, −1.40894899581373596402215818236, −0.842921937877344002769425761715,
0.842921937877344002769425761715, 1.40894899581373596402215818236, 3.11593373105241193282252090259, 4.29234122538525818419634575925, 5.57164932306202968725455985922, 5.88814764717059309422166354874, 6.93134510642987091352839699023, 7.85876996470315743675158079354, 8.575489513983043117314577891579, 9.427346487934332836565213412700
