| L(s) = 1 | + 9.69·3-s − 15·5-s + 9.38·7-s + 66.9·9-s − 46.4·11-s + 51.9·13-s − 145.·15-s + 4.33·17-s − 123.·19-s + 90.9·21-s − 86.2·23-s + 100·25-s + 386.·27-s − 29·29-s − 275.·31-s − 450.·33-s − 140.·35-s + 97.5·37-s + 502.·39-s + 374.·41-s − 436.·43-s − 1.00e3·45-s + 101.·47-s − 255·49-s + 42·51-s − 85.5·53-s + 696.·55-s + ⋯ |
| L(s) = 1 | + 1.86·3-s − 1.34·5-s + 0.506·7-s + 2.47·9-s − 1.27·11-s + 1.10·13-s − 2.50·15-s + 0.0618·17-s − 1.49·19-s + 0.944·21-s − 0.782·23-s + 0.800·25-s + 2.75·27-s − 0.185·29-s − 1.59·31-s − 2.37·33-s − 0.679·35-s + 0.433·37-s + 2.06·39-s + 1.42·41-s − 1.54·43-s − 3.32·45-s + 0.314·47-s − 0.743·49-s + 0.115·51-s − 0.221·53-s + 1.70·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 29 | \( 1 + 29T \) |
| good | 3 | \( 1 - 9.69T + 27T^{2} \) |
| 5 | \( 1 + 15T + 125T^{2} \) |
| 7 | \( 1 - 9.38T + 343T^{2} \) |
| 11 | \( 1 + 46.4T + 1.33e3T^{2} \) |
| 13 | \( 1 - 51.9T + 2.19e3T^{2} \) |
| 17 | \( 1 - 4.33T + 4.91e3T^{2} \) |
| 19 | \( 1 + 123.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 86.2T + 1.21e4T^{2} \) |
| 31 | \( 1 + 275.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 97.5T + 5.06e4T^{2} \) |
| 41 | \( 1 - 374.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 436.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 101.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 85.5T + 1.48e5T^{2} \) |
| 59 | \( 1 + 731.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 526.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 466.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 908.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 250.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 897.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 468.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 314.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 555.T + 9.12e5T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.421431929220061095747680077287, −7.78270981013946121784233856747, −7.52485323201998606170001039482, −6.28499925570036667210139923384, −4.83908953453018392625223780460, −4.00548828812516211246035821926, −3.51935656741539732800919898066, −2.51363815212793162323261835807, −1.61507418249031072808759101334, 0,
1.61507418249031072808759101334, 2.51363815212793162323261835807, 3.51935656741539732800919898066, 4.00548828812516211246035821926, 4.83908953453018392625223780460, 6.28499925570036667210139923384, 7.52485323201998606170001039482, 7.78270981013946121784233856747, 8.421431929220061095747680077287
