| L(s) = 1 | + 3.70·2-s + 3·3-s + 5.70·4-s + 5·5-s + 11.1·6-s − 9.10·7-s − 8.50·8-s + 9·9-s + 18.5·10-s + 17.1·12-s + 20.2·13-s − 33.7·14-s + 15·15-s − 77.1·16-s − 61.8·17-s + 33.3·18-s − 8.68·19-s + 28.5·20-s − 27.3·21-s − 127.·23-s − 25.5·24-s + 25·25-s + 75.1·26-s + 27·27-s − 51.9·28-s − 85.9·29-s + 55.5·30-s + ⋯ |
| L(s) = 1 | + 1.30·2-s + 0.577·3-s + 0.712·4-s + 0.447·5-s + 0.755·6-s − 0.491·7-s − 0.375·8-s + 0.333·9-s + 0.585·10-s + 0.411·12-s + 0.433·13-s − 0.643·14-s + 0.258·15-s − 1.20·16-s − 0.881·17-s + 0.436·18-s − 0.104·19-s + 0.318·20-s − 0.283·21-s − 1.15·23-s − 0.217·24-s + 0.200·25-s + 0.566·26-s + 0.192·27-s − 0.350·28-s − 0.550·29-s + 0.337·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - 3T \) |
| 5 | \( 1 - 5T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 3.70T + 8T^{2} \) |
| 7 | \( 1 + 9.10T + 343T^{2} \) |
| 13 | \( 1 - 20.2T + 2.19e3T^{2} \) |
| 17 | \( 1 + 61.8T + 4.91e3T^{2} \) |
| 19 | \( 1 + 8.68T + 6.85e3T^{2} \) |
| 23 | \( 1 + 127.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 85.9T + 2.43e4T^{2} \) |
| 31 | \( 1 + 158.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 138.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 28.2T + 6.89e4T^{2} \) |
| 43 | \( 1 - 265.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 515.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 466.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 373.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 84.8T + 2.26e5T^{2} \) |
| 67 | \( 1 + 88.7T + 3.00e5T^{2} \) |
| 71 | \( 1 + 536.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 322.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 265.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 520.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 464.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 204.T + 9.12e5T^{2} \) |
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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
−8.598589875853202426457129744490, −7.60242878441684910899767958349, −6.53891211914455939643634306485, −6.09918410394423305947878629641, −5.16260555057876345865275581251, −4.24025502974766166786762098241, −3.58148254422038250591354585898, −2.68098720031604988372088964112, −1.78958960317689192300296613911, 0,
1.78958960317689192300296613911, 2.68098720031604988372088964112, 3.58148254422038250591354585898, 4.24025502974766166786762098241, 5.16260555057876345865275581251, 6.09918410394423305947878629641, 6.53891211914455939643634306485, 7.60242878441684910899767958349, 8.598589875853202426457129744490
