L(s) = 1 | − 0.870·3-s − 0.696·5-s − 2.24·9-s + 2.28·11-s + 3.44·13-s + 0.606·15-s − 1.60·17-s + 19-s − 3.13·23-s − 4.51·25-s + 4.56·27-s + 5.41·29-s − 6.32·31-s − 1.98·33-s + 1.04·37-s − 2.99·39-s − 6.58·41-s + 2.10·43-s + 1.56·45-s + 4.62·47-s + 1.39·51-s + 7.87·53-s − 1.59·55-s − 0.870·57-s − 2.24·59-s − 1.54·61-s − 2.40·65-s + ⋯ |
L(s) = 1 | − 0.502·3-s − 0.311·5-s − 0.747·9-s + 0.688·11-s + 0.955·13-s + 0.156·15-s − 0.389·17-s + 0.229·19-s − 0.652·23-s − 0.902·25-s + 0.878·27-s + 1.00·29-s − 1.13·31-s − 0.345·33-s + 0.172·37-s − 0.480·39-s − 1.02·41-s + 0.321·43-s + 0.232·45-s + 0.674·47-s + 0.195·51-s + 1.08·53-s − 0.214·55-s − 0.115·57-s − 0.292·59-s − 0.197·61-s − 0.297·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7448 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| 19 | \( 1 - T \) |
good | 3 | \( 1 + 0.870T + 3T^{2} \) |
| 5 | \( 1 + 0.696T + 5T^{2} \) |
| 11 | \( 1 - 2.28T + 11T^{2} \) |
| 13 | \( 1 - 3.44T + 13T^{2} \) |
| 17 | \( 1 + 1.60T + 17T^{2} \) |
| 23 | \( 1 + 3.13T + 23T^{2} \) |
| 29 | \( 1 - 5.41T + 29T^{2} \) |
| 31 | \( 1 + 6.32T + 31T^{2} \) |
| 37 | \( 1 - 1.04T + 37T^{2} \) |
| 41 | \( 1 + 6.58T + 41T^{2} \) |
| 43 | \( 1 - 2.10T + 43T^{2} \) |
| 47 | \( 1 - 4.62T + 47T^{2} \) |
| 53 | \( 1 - 7.87T + 53T^{2} \) |
| 59 | \( 1 + 2.24T + 59T^{2} \) |
| 61 | \( 1 + 1.54T + 61T^{2} \) |
| 67 | \( 1 - 0.258T + 67T^{2} \) |
| 71 | \( 1 + 4.57T + 71T^{2} \) |
| 73 | \( 1 + 7.75T + 73T^{2} \) |
| 79 | \( 1 - 8.00T + 79T^{2} \) |
| 83 | \( 1 + 2.52T + 83T^{2} \) |
| 89 | \( 1 + 16.2T + 89T^{2} \) |
| 97 | \( 1 - 8.33T + 97T^{2} \) |
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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
−7.49441404377308122902069613531, −6.77513709519353970437799818489, −6.02694990489674420052492504386, −5.67033351399103444176876108594, −4.68095405399820866962912935894, −3.92727742627857200186417511241, −3.28289688513470498427015241846, −2.20908505881445596127725398601, −1.15014433449769652272896331208, 0,
1.15014433449769652272896331208, 2.20908505881445596127725398601, 3.28289688513470498427015241846, 3.92727742627857200186417511241, 4.68095405399820866962912935894, 5.67033351399103444176876108594, 6.02694990489674420052492504386, 6.77513709519353970437799818489, 7.49441404377308122902069613531