| L(s) = 1 | − 2-s + 1.28·3-s + 4-s − 3.54·5-s − 1.28·6-s + 3.93·7-s − 8-s − 1.34·9-s + 3.54·10-s − 6.17·11-s + 1.28·12-s + 2.81·13-s − 3.93·14-s − 4.56·15-s + 16-s − 1.43·17-s + 1.34·18-s − 2.37·19-s − 3.54·20-s + 5.06·21-s + 6.17·22-s + 8.48·23-s − 1.28·24-s + 7.59·25-s − 2.81·26-s − 5.59·27-s + 3.93·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.742·3-s + 0.5·4-s − 1.58·5-s − 0.525·6-s + 1.48·7-s − 0.353·8-s − 0.448·9-s + 1.12·10-s − 1.86·11-s + 0.371·12-s + 0.781·13-s − 1.05·14-s − 1.17·15-s + 0.250·16-s − 0.347·17-s + 0.316·18-s − 0.544·19-s − 0.793·20-s + 1.10·21-s + 1.31·22-s + 1.76·23-s − 0.262·24-s + 1.51·25-s − 0.552·26-s − 1.07·27-s + 0.744·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1502 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1502 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.162248546\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.162248546\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 751 | \( 1 - T \) |
| good | 3 | \( 1 - 1.28T + 3T^{2} \) |
| 5 | \( 1 + 3.54T + 5T^{2} \) |
| 7 | \( 1 - 3.93T + 7T^{2} \) |
| 11 | \( 1 + 6.17T + 11T^{2} \) |
| 13 | \( 1 - 2.81T + 13T^{2} \) |
| 17 | \( 1 + 1.43T + 17T^{2} \) |
| 19 | \( 1 + 2.37T + 19T^{2} \) |
| 23 | \( 1 - 8.48T + 23T^{2} \) |
| 29 | \( 1 - 10.4T + 29T^{2} \) |
| 31 | \( 1 - 6.97T + 31T^{2} \) |
| 37 | \( 1 - 6.13T + 37T^{2} \) |
| 41 | \( 1 - 6.16T + 41T^{2} \) |
| 43 | \( 1 + 5.66T + 43T^{2} \) |
| 47 | \( 1 - 6.45T + 47T^{2} \) |
| 53 | \( 1 + 10.4T + 53T^{2} \) |
| 59 | \( 1 - 0.145T + 59T^{2} \) |
| 61 | \( 1 + 10.1T + 61T^{2} \) |
| 67 | \( 1 - 5.30T + 67T^{2} \) |
| 71 | \( 1 - 8.17T + 71T^{2} \) |
| 73 | \( 1 - 2.50T + 73T^{2} \) |
| 79 | \( 1 - 7.09T + 79T^{2} \) |
| 83 | \( 1 - 11.4T + 83T^{2} \) |
| 89 | \( 1 - 11.1T + 89T^{2} \) |
| 97 | \( 1 + 4.24T + 97T^{2} \) |
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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
−9.073887230482708142692093857849, −8.400333309086854884694157906136, −7.953125229595768743543414775852, −7.74629589928759382712130472380, −6.51074592389027518814630559769, −5.08644065531209038649392118248, −4.46353796580279304051300983065, −3.14786994109439210140140933373, −2.46716928370755926146553043863, −0.817137777591282559989264115180,
0.817137777591282559989264115180, 2.46716928370755926146553043863, 3.14786994109439210140140933373, 4.46353796580279304051300983065, 5.08644065531209038649392118248, 6.51074592389027518814630559769, 7.74629589928759382712130472380, 7.953125229595768743543414775852, 8.400333309086854884694157906136, 9.073887230482708142692093857849
