L(s) = 1 | − 2-s + 0.812·3-s + 4-s + 0.413·5-s − 0.812·6-s − 0.531·7-s − 8-s − 2.33·9-s − 0.413·10-s + 5.43·11-s + 0.812·12-s + 0.828·13-s + 0.531·14-s + 0.336·15-s + 16-s + 1.00·17-s + 2.33·18-s − 6.05·19-s + 0.413·20-s − 0.431·21-s − 5.43·22-s + 23-s − 0.812·24-s − 4.82·25-s − 0.828·26-s − 4.34·27-s − 0.531·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.469·3-s + 0.5·4-s + 0.184·5-s − 0.331·6-s − 0.200·7-s − 0.353·8-s − 0.779·9-s − 0.130·10-s + 1.63·11-s + 0.234·12-s + 0.229·13-s + 0.141·14-s + 0.0868·15-s + 0.250·16-s + 0.244·17-s + 0.551·18-s − 1.38·19-s + 0.0924·20-s − 0.0942·21-s − 1.15·22-s + 0.208·23-s − 0.165·24-s − 0.965·25-s − 0.162·26-s − 0.835·27-s − 0.100·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.622977629\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.622977629\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| 131 | \( 1 + T \) |
good | 3 | \( 1 - 0.812T + 3T^{2} \) |
| 5 | \( 1 - 0.413T + 5T^{2} \) |
| 7 | \( 1 + 0.531T + 7T^{2} \) |
| 11 | \( 1 - 5.43T + 11T^{2} \) |
| 13 | \( 1 - 0.828T + 13T^{2} \) |
| 17 | \( 1 - 1.00T + 17T^{2} \) |
| 19 | \( 1 + 6.05T + 19T^{2} \) |
| 29 | \( 1 + 0.830T + 29T^{2} \) |
| 31 | \( 1 - 3.41T + 31T^{2} \) |
| 37 | \( 1 + 4.77T + 37T^{2} \) |
| 41 | \( 1 - 8.06T + 41T^{2} \) |
| 43 | \( 1 + 0.794T + 43T^{2} \) |
| 47 | \( 1 - 8.99T + 47T^{2} \) |
| 53 | \( 1 - 13.4T + 53T^{2} \) |
| 59 | \( 1 - 4.21T + 59T^{2} \) |
| 61 | \( 1 - 4.62T + 61T^{2} \) |
| 67 | \( 1 - 2.68T + 67T^{2} \) |
| 71 | \( 1 - 7.56T + 71T^{2} \) |
| 73 | \( 1 - 12.5T + 73T^{2} \) |
| 79 | \( 1 + 1.38T + 79T^{2} \) |
| 83 | \( 1 + 2.61T + 83T^{2} \) |
| 89 | \( 1 + 11.3T + 89T^{2} \) |
| 97 | \( 1 - 4.82T + 97T^{2} \) |
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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
−8.367063714728764013591123917414, −7.46015578274434672344084497183, −6.65530946529555012728453486676, −6.16930706027663046175243016902, −5.44223798810513339864777574198, −4.11701671169891286318511606762, −3.65795892287988269846685377478, −2.57071363154626288361354426577, −1.86880084368889407156554917377, −0.72524751782374523276315915576,
0.72524751782374523276315915576, 1.86880084368889407156554917377, 2.57071363154626288361354426577, 3.65795892287988269846685377478, 4.11701671169891286318511606762, 5.44223798810513339864777574198, 6.16930706027663046175243016902, 6.65530946529555012728453486676, 7.46015578274434672344084497183, 8.367063714728764013591123917414