L(s) = 1 | + 2-s + 4-s − 0.107·5-s − 0.389·7-s + 8-s − 0.107·10-s − 3.56·11-s + 0.940·13-s − 0.389·14-s + 16-s + 5.00·17-s − 2.96·19-s − 0.107·20-s − 3.56·22-s + 5.72·23-s − 4.98·25-s + 0.940·26-s − 0.389·28-s + 3.43·29-s + 10.7·31-s + 32-s + 5.00·34-s + 0.0418·35-s + 2.36·37-s − 2.96·38-s − 0.107·40-s − 7.33·41-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.5·4-s − 0.0480·5-s − 0.147·7-s + 0.353·8-s − 0.0339·10-s − 1.07·11-s + 0.260·13-s − 0.104·14-s + 0.250·16-s + 1.21·17-s − 0.679·19-s − 0.0240·20-s − 0.760·22-s + 1.19·23-s − 0.997·25-s + 0.184·26-s − 0.0736·28-s + 0.637·29-s + 1.92·31-s + 0.176·32-s + 0.857·34-s + 0.00707·35-s + 0.389·37-s − 0.480·38-s − 0.0169·40-s − 1.14·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6354 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.040465501\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.040465501\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 353 | \( 1 - T \) |
good | 5 | \( 1 + 0.107T + 5T^{2} \) |
| 7 | \( 1 + 0.389T + 7T^{2} \) |
| 11 | \( 1 + 3.56T + 11T^{2} \) |
| 13 | \( 1 - 0.940T + 13T^{2} \) |
| 17 | \( 1 - 5.00T + 17T^{2} \) |
| 19 | \( 1 + 2.96T + 19T^{2} \) |
| 23 | \( 1 - 5.72T + 23T^{2} \) |
| 29 | \( 1 - 3.43T + 29T^{2} \) |
| 31 | \( 1 - 10.7T + 31T^{2} \) |
| 37 | \( 1 - 2.36T + 37T^{2} \) |
| 41 | \( 1 + 7.33T + 41T^{2} \) |
| 43 | \( 1 + 2.88T + 43T^{2} \) |
| 47 | \( 1 - 1.58T + 47T^{2} \) |
| 53 | \( 1 + 9.08T + 53T^{2} \) |
| 59 | \( 1 - 14.7T + 59T^{2} \) |
| 61 | \( 1 - 8.54T + 61T^{2} \) |
| 67 | \( 1 + 2.03T + 67T^{2} \) |
| 71 | \( 1 - 0.426T + 71T^{2} \) |
| 73 | \( 1 + 4.96T + 73T^{2} \) |
| 79 | \( 1 + 3.04T + 79T^{2} \) |
| 83 | \( 1 - 12.9T + 83T^{2} \) |
| 89 | \( 1 - 8.76T + 89T^{2} \) |
| 97 | \( 1 + 1.17T + 97T^{2} \) |
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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
−8.111185473912795816075649159122, −7.23588150953279303980730725734, −6.50354395705934543654520901755, −5.86700926324367404074151499646, −5.07330451133528725674682237595, −4.57741564005568698721470732813, −3.49887045478235963701242240867, −2.95935023212851383953438732272, −2.04482534678921077156980327641, −0.812366422559584226852947110233,
0.812366422559584226852947110233, 2.04482534678921077156980327641, 2.95935023212851383953438732272, 3.49887045478235963701242240867, 4.57741564005568698721470732813, 5.07330451133528725674682237595, 5.86700926324367404074151499646, 6.50354395705934543654520901755, 7.23588150953279303980730725734, 8.111185473912795816075649159122