| L(s) = 1 | − 3-s − 5-s − 1.35·7-s + 9-s − 3.58·11-s + 13-s + 15-s − 7.81·17-s − 4.94·19-s + 1.35·21-s − 0.0737·23-s + 25-s − 27-s + 5.51·29-s − 4·31-s + 3.58·33-s + 1.35·35-s − 1.58·37-s − 39-s − 5.58·41-s − 7.17·43-s − 45-s − 2.56·47-s − 5.15·49-s + 7.81·51-s + 12.7·53-s + 3.58·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 0.513·7-s + 0.333·9-s − 1.08·11-s + 0.277·13-s + 0.258·15-s − 1.89·17-s − 1.13·19-s + 0.296·21-s − 0.0153·23-s + 0.200·25-s − 0.192·27-s + 1.02·29-s − 0.718·31-s + 0.624·33-s + 0.229·35-s − 0.261·37-s − 0.160·39-s − 0.872·41-s − 1.09·43-s − 0.149·45-s − 0.374·47-s − 0.736·49-s + 1.09·51-s + 1.75·53-s + 0.483·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4532106112\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4532106112\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 + 1.35T + 7T^{2} \) |
| 11 | \( 1 + 3.58T + 11T^{2} \) |
| 17 | \( 1 + 7.81T + 17T^{2} \) |
| 19 | \( 1 + 4.94T + 19T^{2} \) |
| 23 | \( 1 + 0.0737T + 23T^{2} \) |
| 29 | \( 1 - 5.51T + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 + 1.58T + 37T^{2} \) |
| 41 | \( 1 + 5.58T + 41T^{2} \) |
| 43 | \( 1 + 7.17T + 43T^{2} \) |
| 47 | \( 1 + 2.56T + 47T^{2} \) |
| 53 | \( 1 - 12.7T + 53T^{2} \) |
| 59 | \( 1 - 8.45T + 59T^{2} \) |
| 61 | \( 1 + 4.30T + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 - 0.871T + 71T^{2} \) |
| 73 | \( 1 - 6.79T + 73T^{2} \) |
| 79 | \( 1 + 12.8T + 79T^{2} \) |
| 83 | \( 1 - 5.43T + 83T^{2} \) |
| 89 | \( 1 + 6.87T + 89T^{2} \) |
| 97 | \( 1 - 7.35T + 97T^{2} \) |
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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
−8.264454503209869781228117862104, −7.03736772442931028999836293926, −6.76394350716210226102843268728, −5.99294247676327140350225776342, −5.11317096441213363646525872820, −4.50141902085772549251461740741, −3.73739995862293131032922008528, −2.73324285196159800349578383176, −1.88980673205517871866056331925, −0.34177022352854767219290519722,
0.34177022352854767219290519722, 1.88980673205517871866056331925, 2.73324285196159800349578383176, 3.73739995862293131032922008528, 4.50141902085772549251461740741, 5.11317096441213363646525872820, 5.99294247676327140350225776342, 6.76394350716210226102843268728, 7.03736772442931028999836293926, 8.264454503209869781228117862104
