L(s) = 1 | − 5·5-s − 12·7-s + 24·11-s + 38·13-s + 6·17-s + 104·19-s − 100·23-s + 25·25-s − 230·29-s − 56·31-s + 60·35-s + 190·37-s − 202·41-s − 148·43-s − 124·47-s − 199·49-s − 206·53-s − 120·55-s + 128·59-s + 190·61-s − 190·65-s − 204·67-s + 440·71-s + 1.21e3·73-s − 288·77-s + 816·79-s + 1.41e3·83-s + ⋯ |
L(s) = 1 | − 0.447·5-s − 0.647·7-s + 0.657·11-s + 0.810·13-s + 0.0856·17-s + 1.25·19-s − 0.906·23-s + 1/5·25-s − 1.47·29-s − 0.324·31-s + 0.289·35-s + 0.844·37-s − 0.769·41-s − 0.524·43-s − 0.384·47-s − 0.580·49-s − 0.533·53-s − 0.294·55-s + 0.282·59-s + 0.398·61-s − 0.362·65-s − 0.371·67-s + 0.735·71-s + 1.93·73-s − 0.426·77-s + 1.16·79-s + 1.86·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.720260743\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.720260743\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + p T \) |
good | 7 | \( 1 + 12 T + p^{3} T^{2} \) |
| 11 | \( 1 - 24 T + p^{3} T^{2} \) |
| 13 | \( 1 - 38 T + p^{3} T^{2} \) |
| 17 | \( 1 - 6 T + p^{3} T^{2} \) |
| 19 | \( 1 - 104 T + p^{3} T^{2} \) |
| 23 | \( 1 + 100 T + p^{3} T^{2} \) |
| 29 | \( 1 + 230 T + p^{3} T^{2} \) |
| 31 | \( 1 + 56 T + p^{3} T^{2} \) |
| 37 | \( 1 - 190 T + p^{3} T^{2} \) |
| 41 | \( 1 + 202 T + p^{3} T^{2} \) |
| 43 | \( 1 + 148 T + p^{3} T^{2} \) |
| 47 | \( 1 + 124 T + p^{3} T^{2} \) |
| 53 | \( 1 + 206 T + p^{3} T^{2} \) |
| 59 | \( 1 - 128 T + p^{3} T^{2} \) |
| 61 | \( 1 - 190 T + p^{3} T^{2} \) |
| 67 | \( 1 + 204 T + p^{3} T^{2} \) |
| 71 | \( 1 - 440 T + p^{3} T^{2} \) |
| 73 | \( 1 - 1210 T + p^{3} T^{2} \) |
| 79 | \( 1 - 816 T + p^{3} T^{2} \) |
| 83 | \( 1 - 1412 T + p^{3} T^{2} \) |
| 89 | \( 1 - 214 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1202 T + p^{3} T^{2} \) |
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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
−9.342291157583155289271504499542, −8.297240907474522208917927019625, −7.59980455529855389380735656028, −6.67473525946051834617650169888, −5.98095234744875611285998651900, −4.99696382085790261209211444229, −3.75534862255638371370750616737, −3.37542314597662364538924813505, −1.87311514469507278264387557306, −0.65128941001997020951646049991,
0.65128941001997020951646049991, 1.87311514469507278264387557306, 3.37542314597662364538924813505, 3.75534862255638371370750616737, 4.99696382085790261209211444229, 5.98095234744875611285998651900, 6.67473525946051834617650169888, 7.59980455529855389380735656028, 8.297240907474522208917927019625, 9.342291157583155289271504499542