L(s) = 1 | − 2-s − 0.141·3-s + 4-s − 1.57·5-s + 0.141·6-s + 7-s − 8-s − 2.98·9-s + 1.57·10-s − 0.868·11-s − 0.141·12-s − 4.18·13-s − 14-s + 0.222·15-s + 16-s + 5.52·17-s + 2.98·18-s + 0.527·19-s − 1.57·20-s − 0.141·21-s + 0.868·22-s + 2.36·23-s + 0.141·24-s − 2.51·25-s + 4.18·26-s + 0.844·27-s + 28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.0814·3-s + 0.5·4-s − 0.705·5-s + 0.0576·6-s + 0.377·7-s − 0.353·8-s − 0.993·9-s + 0.498·10-s − 0.261·11-s − 0.0407·12-s − 1.16·13-s − 0.267·14-s + 0.0574·15-s + 0.250·16-s + 1.33·17-s + 0.702·18-s + 0.121·19-s − 0.352·20-s − 0.0308·21-s + 0.185·22-s + 0.492·23-s + 0.0288·24-s − 0.502·25-s + 0.821·26-s + 0.162·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 431 | \( 1 - T \) |
good | 3 | \( 1 + 0.141T + 3T^{2} \) |
| 5 | \( 1 + 1.57T + 5T^{2} \) |
| 11 | \( 1 + 0.868T + 11T^{2} \) |
| 13 | \( 1 + 4.18T + 13T^{2} \) |
| 17 | \( 1 - 5.52T + 17T^{2} \) |
| 19 | \( 1 - 0.527T + 19T^{2} \) |
| 23 | \( 1 - 2.36T + 23T^{2} \) |
| 29 | \( 1 - 6.24T + 29T^{2} \) |
| 31 | \( 1 - 4.35T + 31T^{2} \) |
| 37 | \( 1 + 8.73T + 37T^{2} \) |
| 41 | \( 1 + 10.6T + 41T^{2} \) |
| 43 | \( 1 - 8.28T + 43T^{2} \) |
| 47 | \( 1 - 9.80T + 47T^{2} \) |
| 53 | \( 1 + 0.274T + 53T^{2} \) |
| 59 | \( 1 + 4.14T + 59T^{2} \) |
| 61 | \( 1 - 11.0T + 61T^{2} \) |
| 67 | \( 1 - 4.92T + 67T^{2} \) |
| 71 | \( 1 - 3.02T + 71T^{2} \) |
| 73 | \( 1 + 1.36T + 73T^{2} \) |
| 79 | \( 1 - 5.42T + 79T^{2} \) |
| 83 | \( 1 + 1.60T + 83T^{2} \) |
| 89 | \( 1 + 13.8T + 89T^{2} \) |
| 97 | \( 1 + 8.09T + 97T^{2} \) |
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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
−7.82173572464990354996539951348, −7.25337606675783717563536553904, −6.48247829534948633852645565499, −5.43822306624176880723410095650, −5.08023914428669804687829488444, −3.93207189114528679839043385072, −3.03420552594379362764905479485, −2.37601095419211137002815724524, −1.06342741406383541760072929990, 0,
1.06342741406383541760072929990, 2.37601095419211137002815724524, 3.03420552594379362764905479485, 3.93207189114528679839043385072, 5.08023914428669804687829488444, 5.43822306624176880723410095650, 6.48247829534948633852645565499, 7.25337606675783717563536553904, 7.82173572464990354996539951348