L(s) = 1 | + 8.77·3-s + 17.3·5-s + 26.0·7-s + 49.9·9-s − 4.22·11-s − 64.0·13-s + 151.·15-s − 48.5·17-s + 19·19-s + 228.·21-s − 92.0·23-s + 174.·25-s + 201.·27-s + 88.2·29-s + 81.9·31-s − 37.0·33-s + 451.·35-s + 23.6·37-s − 561.·39-s + 17.7·41-s + 368.·43-s + 864.·45-s + 497.·47-s + 337.·49-s − 425.·51-s + 536.·53-s − 73.2·55-s + ⋯ |
L(s) = 1 | + 1.68·3-s + 1.54·5-s + 1.40·7-s + 1.84·9-s − 0.115·11-s − 1.36·13-s + 2.61·15-s − 0.692·17-s + 0.229·19-s + 2.37·21-s − 0.834·23-s + 1.39·25-s + 1.43·27-s + 0.564·29-s + 0.474·31-s − 0.195·33-s + 2.18·35-s + 0.104·37-s − 2.30·39-s + 0.0674·41-s + 1.30·43-s + 2.86·45-s + 1.54·47-s + 0.984·49-s − 1.16·51-s + 1.39·53-s − 0.179·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(6.288734471\) |
\(L(\frac12)\) |
\(\approx\) |
\(6.288734471\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 - 19T \) |
good | 3 | \( 1 - 8.77T + 27T^{2} \) |
| 5 | \( 1 - 17.3T + 125T^{2} \) |
| 7 | \( 1 - 26.0T + 343T^{2} \) |
| 11 | \( 1 + 4.22T + 1.33e3T^{2} \) |
| 13 | \( 1 + 64.0T + 2.19e3T^{2} \) |
| 17 | \( 1 + 48.5T + 4.91e3T^{2} \) |
| 23 | \( 1 + 92.0T + 1.21e4T^{2} \) |
| 29 | \( 1 - 88.2T + 2.43e4T^{2} \) |
| 31 | \( 1 - 81.9T + 2.97e4T^{2} \) |
| 37 | \( 1 - 23.6T + 5.06e4T^{2} \) |
| 41 | \( 1 - 17.7T + 6.89e4T^{2} \) |
| 43 | \( 1 - 368.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 497.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 536.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 36.6T + 2.05e5T^{2} \) |
| 61 | \( 1 + 630.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 282.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 595.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 597.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 427.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 493.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 921.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.08e3T + 9.12e5T^{2} \) |
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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
−9.185973526962627826523905023070, −8.718663720873393263019731339468, −7.78582264149352925714818608840, −7.22620626873839103528204963063, −5.96159475927526350770924275753, −4.94639892186479683212488840222, −4.19435562168055639873312091549, −2.57393821642082024247090844101, −2.30606826442861438478680056216, −1.36612815329328445351625252446,
1.36612815329328445351625252446, 2.30606826442861438478680056216, 2.57393821642082024247090844101, 4.19435562168055639873312091549, 4.94639892186479683212488840222, 5.96159475927526350770924275753, 7.22620626873839103528204963063, 7.78582264149352925714818608840, 8.718663720873393263019731339468, 9.185973526962627826523905023070