L(s) = 1 | + 2-s + 0.618·3-s + 4-s + 5-s + 0.618·6-s − 2.61·7-s + 8-s − 2.61·9-s + 10-s + 11-s + 0.618·12-s + 4.47·13-s − 2.61·14-s + 0.618·15-s + 16-s − 6.09·17-s − 2.61·18-s − 2.38·19-s + 20-s − 1.61·21-s + 22-s − 4·23-s + 0.618·24-s + 25-s + 4.47·26-s − 3.47·27-s − 2.61·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.356·3-s + 0.5·4-s + 0.447·5-s + 0.252·6-s − 0.989·7-s + 0.353·8-s − 0.872·9-s + 0.316·10-s + 0.301·11-s + 0.178·12-s + 1.24·13-s − 0.699·14-s + 0.159·15-s + 0.250·16-s − 1.47·17-s − 0.617·18-s − 0.546·19-s + 0.223·20-s − 0.353·21-s + 0.213·22-s − 0.834·23-s + 0.126·24-s + 0.200·25-s + 0.877·26-s − 0.668·27-s − 0.494·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4730 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4730 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 43 | \( 1 - T \) |
good | 3 | \( 1 - 0.618T + 3T^{2} \) |
| 7 | \( 1 + 2.61T + 7T^{2} \) |
| 13 | \( 1 - 4.47T + 13T^{2} \) |
| 17 | \( 1 + 6.09T + 17T^{2} \) |
| 19 | \( 1 + 2.38T + 19T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 + 4.47T + 29T^{2} \) |
| 31 | \( 1 + 2.47T + 31T^{2} \) |
| 37 | \( 1 + 3.23T + 37T^{2} \) |
| 41 | \( 1 + 10T + 41T^{2} \) |
| 47 | \( 1 + 5.38T + 47T^{2} \) |
| 53 | \( 1 - 4.09T + 53T^{2} \) |
| 59 | \( 1 - 5.38T + 59T^{2} \) |
| 61 | \( 1 - 2.47T + 61T^{2} \) |
| 67 | \( 1 + 8T + 67T^{2} \) |
| 71 | \( 1 - 7.09T + 71T^{2} \) |
| 73 | \( 1 - 4.47T + 73T^{2} \) |
| 79 | \( 1 + 7.09T + 79T^{2} \) |
| 83 | \( 1 - 17.5T + 83T^{2} \) |
| 89 | \( 1 - 8.47T + 89T^{2} \) |
| 97 | \( 1 + 7.23T + 97T^{2} \) |
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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
−8.020478243187404153971826766538, −6.83702342770836978655298901372, −6.41745676027085323571113210525, −5.86441223724595726496522245597, −5.03841696150835861194841701527, −3.87868102511436764445371617241, −3.52059305225043462167506677433, −2.51120741808663230297406935429, −1.76011059079985019230736554771, 0,
1.76011059079985019230736554771, 2.51120741808663230297406935429, 3.52059305225043462167506677433, 3.87868102511436764445371617241, 5.03841696150835861194841701527, 5.86441223724595726496522245597, 6.41745676027085323571113210525, 6.83702342770836978655298901372, 8.020478243187404153971826766538