| L(s) = 1 | − 2-s − 0.593·3-s + 4-s + 5-s + 0.593·6-s + 4.51·7-s − 8-s − 2.64·9-s − 10-s + 1.40·11-s − 0.593·12-s − 6.32·13-s − 4.51·14-s − 0.593·15-s + 16-s + 2.05·17-s + 2.64·18-s − 4.46·19-s + 20-s − 2.67·21-s − 1.40·22-s + 5.51·23-s + 0.593·24-s + 25-s + 6.32·26-s + 3.35·27-s + 4.51·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.342·3-s + 0.5·4-s + 0.447·5-s + 0.242·6-s + 1.70·7-s − 0.353·8-s − 0.882·9-s − 0.316·10-s + 0.424·11-s − 0.171·12-s − 1.75·13-s − 1.20·14-s − 0.153·15-s + 0.250·16-s + 0.498·17-s + 0.624·18-s − 1.02·19-s + 0.223·20-s − 0.584·21-s − 0.299·22-s + 1.14·23-s + 0.121·24-s + 0.200·25-s + 1.24·26-s + 0.645·27-s + 0.853·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9610 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9610 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.397784443\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.397784443\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 31 | \( 1 \) |
| good | 3 | \( 1 + 0.593T + 3T^{2} \) |
| 7 | \( 1 - 4.51T + 7T^{2} \) |
| 11 | \( 1 - 1.40T + 11T^{2} \) |
| 13 | \( 1 + 6.32T + 13T^{2} \) |
| 17 | \( 1 - 2.05T + 17T^{2} \) |
| 19 | \( 1 + 4.46T + 19T^{2} \) |
| 23 | \( 1 - 5.51T + 23T^{2} \) |
| 29 | \( 1 + 1.32T + 29T^{2} \) |
| 37 | \( 1 - 4.32T + 37T^{2} \) |
| 41 | \( 1 + 12.1T + 41T^{2} \) |
| 43 | \( 1 - 2.46T + 43T^{2} \) |
| 47 | \( 1 + 4.38T + 47T^{2} \) |
| 53 | \( 1 - 4.86T + 53T^{2} \) |
| 59 | \( 1 + 13.1T + 59T^{2} \) |
| 61 | \( 1 - 0.891T + 61T^{2} \) |
| 67 | \( 1 + 1.37T + 67T^{2} \) |
| 71 | \( 1 - 6.43T + 71T^{2} \) |
| 73 | \( 1 - 6.38T + 73T^{2} \) |
| 79 | \( 1 - 6.32T + 79T^{2} \) |
| 83 | \( 1 - 14.3T + 83T^{2} \) |
| 89 | \( 1 - 2.72T + 89T^{2} \) |
| 97 | \( 1 + 5.32T + 97T^{2} \) |
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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
−7.88376535277291617629802008542, −7.06342225885206576687340222261, −6.46451923297649950990820286324, −5.52160094273289584505844810303, −5.04447147858500260288430190317, −4.49693896397925506023444034312, −3.18395918701216956804597845260, −2.30169293783595763353235163495, −1.72603852848213864734430902772, −0.63823361404291572569200802238,
0.63823361404291572569200802238, 1.72603852848213864734430902772, 2.30169293783595763353235163495, 3.18395918701216956804597845260, 4.49693896397925506023444034312, 5.04447147858500260288430190317, 5.52160094273289584505844810303, 6.46451923297649950990820286324, 7.06342225885206576687340222261, 7.88376535277291617629802008542
