L(s) = 1 | + 2-s + 3-s + 4-s − 0.600·5-s + 6-s + 0.748·7-s + 8-s + 9-s − 0.600·10-s + 11-s + 12-s − 3.37·13-s + 0.748·14-s − 0.600·15-s + 16-s + 1.61·17-s + 18-s + 1.26·19-s − 0.600·20-s + 0.748·21-s + 22-s + 5.33·23-s + 24-s − 4.63·25-s − 3.37·26-s + 27-s + 0.748·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.268·5-s + 0.408·6-s + 0.282·7-s + 0.353·8-s + 0.333·9-s − 0.189·10-s + 0.301·11-s + 0.288·12-s − 0.935·13-s + 0.199·14-s − 0.155·15-s + 0.250·16-s + 0.392·17-s + 0.235·18-s + 0.289·19-s − 0.134·20-s + 0.163·21-s + 0.213·22-s + 1.11·23-s + 0.204·24-s − 0.927·25-s − 0.661·26-s + 0.192·27-s + 0.141·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.919915998\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.919915998\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 61 | \( 1 + T \) |
good | 5 | \( 1 + 0.600T + 5T^{2} \) |
| 7 | \( 1 - 0.748T + 7T^{2} \) |
| 13 | \( 1 + 3.37T + 13T^{2} \) |
| 17 | \( 1 - 1.61T + 17T^{2} \) |
| 19 | \( 1 - 1.26T + 19T^{2} \) |
| 23 | \( 1 - 5.33T + 23T^{2} \) |
| 29 | \( 1 - 3.93T + 29T^{2} \) |
| 31 | \( 1 - 6.73T + 31T^{2} \) |
| 37 | \( 1 - 0.919T + 37T^{2} \) |
| 41 | \( 1 - 10.3T + 41T^{2} \) |
| 43 | \( 1 - 0.371T + 43T^{2} \) |
| 47 | \( 1 - 1.78T + 47T^{2} \) |
| 53 | \( 1 + 8.73T + 53T^{2} \) |
| 59 | \( 1 + 1.55T + 59T^{2} \) |
| 67 | \( 1 - 6.96T + 67T^{2} \) |
| 71 | \( 1 - 7.20T + 71T^{2} \) |
| 73 | \( 1 - 11.0T + 73T^{2} \) |
| 79 | \( 1 - 15.8T + 79T^{2} \) |
| 83 | \( 1 - 5.44T + 83T^{2} \) |
| 89 | \( 1 - 10.6T + 89T^{2} \) |
| 97 | \( 1 + 17.4T + 97T^{2} \) |
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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
−8.070833752057286891239639924251, −7.86817487864206581217362484327, −6.93682435621422280690934342027, −6.30265873975693758915474329475, −5.23229366443099064572943681086, −4.66239516507539799319949507883, −3.85361825472067135642379591180, −2.99323337233510096108083679797, −2.26894028276143987109713952680, −1.03890674944333899940515016160,
1.03890674944333899940515016160, 2.26894028276143987109713952680, 2.99323337233510096108083679797, 3.85361825472067135642379591180, 4.66239516507539799319949507883, 5.23229366443099064572943681086, 6.30265873975693758915474329475, 6.93682435621422280690934342027, 7.86817487864206581217362484327, 8.070833752057286891239639924251