| L(s) = 1 | + 1.58·2-s − 3·3-s − 5.49·4-s + 1.38·5-s − 4.74·6-s − 21.3·8-s + 9·9-s + 2.19·10-s + 24.7·11-s + 16.4·12-s + 13·13-s − 4.16·15-s + 10.1·16-s + 126.·17-s + 14.2·18-s + 82.8·19-s − 7.62·20-s + 39.2·22-s − 67.3·23-s + 64.0·24-s − 123.·25-s + 20.5·26-s − 27·27-s − 228.·29-s − 6.59·30-s − 101.·31-s + 186.·32-s + ⋯ |
| L(s) = 1 | + 0.559·2-s − 0.577·3-s − 0.686·4-s + 0.124·5-s − 0.323·6-s − 0.944·8-s + 0.333·9-s + 0.0694·10-s + 0.679·11-s + 0.396·12-s + 0.277·13-s − 0.0716·15-s + 0.158·16-s + 1.80·17-s + 0.186·18-s + 1.00·19-s − 0.0852·20-s + 0.380·22-s − 0.610·23-s + 0.545·24-s − 0.984·25-s + 0.155·26-s − 0.192·27-s − 1.46·29-s − 0.0401·30-s − 0.586·31-s + 1.03·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.837765618\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.837765618\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3T \) |
| 7 | \( 1 \) |
| 13 | \( 1 - 13T \) |
| good | 2 | \( 1 - 1.58T + 8T^{2} \) |
| 5 | \( 1 - 1.38T + 125T^{2} \) |
| 11 | \( 1 - 24.7T + 1.33e3T^{2} \) |
| 17 | \( 1 - 126.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 82.8T + 6.85e3T^{2} \) |
| 23 | \( 1 + 67.3T + 1.21e4T^{2} \) |
| 29 | \( 1 + 228.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 101.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 188.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 198.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 283.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 157.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 7.59T + 1.48e5T^{2} \) |
| 59 | \( 1 + 229.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 630.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 45.9T + 3.00e5T^{2} \) |
| 71 | \( 1 - 440.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 624.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 55.3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 782.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.48e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.79e3T + 9.12e5T^{2} \) |
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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
−9.056321849985563958972504139852, −7.938977044408169718674309217183, −7.29740606884354544884488469771, −6.02090258876151115396179249555, −5.68889236444539654042881260926, −4.90747479108265822899630016034, −3.79647308798386831438939961472, −3.39621706456373784932037373592, −1.70434828845081433106645269823, −0.62142429161282153159575950116,
0.62142429161282153159575950116, 1.70434828845081433106645269823, 3.39621706456373784932037373592, 3.79647308798386831438939961472, 4.90747479108265822899630016034, 5.68889236444539654042881260926, 6.02090258876151115396179249555, 7.29740606884354544884488469771, 7.938977044408169718674309217183, 9.056321849985563958972504139852
