L(s) = 1 | − 2.44·3-s − 5-s + 3.62·7-s + 2.98·9-s + 6.10·11-s + 4.48·13-s + 2.44·15-s − 2.11·17-s + 4.61·19-s − 8.88·21-s + 6.89·23-s + 25-s + 0.0292·27-s + 5.25·29-s − 5.03·31-s − 14.9·33-s − 3.62·35-s + 37-s − 10.9·39-s − 10.0·41-s + 5.28·43-s − 2.98·45-s + 1.98·47-s + 6.17·49-s + 5.18·51-s − 4.20·53-s − 6.10·55-s + ⋯ |
L(s) = 1 | − 1.41·3-s − 0.447·5-s + 1.37·7-s + 0.996·9-s + 1.84·11-s + 1.24·13-s + 0.631·15-s − 0.513·17-s + 1.05·19-s − 1.93·21-s + 1.43·23-s + 0.200·25-s + 0.00562·27-s + 0.976·29-s − 0.904·31-s − 2.60·33-s − 0.613·35-s + 0.164·37-s − 1.75·39-s − 1.56·41-s + 0.805·43-s − 0.445·45-s + 0.288·47-s + 0.882·49-s + 0.725·51-s − 0.577·53-s − 0.823·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.585997200\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.585997200\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 37 | \( 1 - T \) |
good | 3 | \( 1 + 2.44T + 3T^{2} \) |
| 7 | \( 1 - 3.62T + 7T^{2} \) |
| 11 | \( 1 - 6.10T + 11T^{2} \) |
| 13 | \( 1 - 4.48T + 13T^{2} \) |
| 17 | \( 1 + 2.11T + 17T^{2} \) |
| 19 | \( 1 - 4.61T + 19T^{2} \) |
| 23 | \( 1 - 6.89T + 23T^{2} \) |
| 29 | \( 1 - 5.25T + 29T^{2} \) |
| 31 | \( 1 + 5.03T + 31T^{2} \) |
| 41 | \( 1 + 10.0T + 41T^{2} \) |
| 43 | \( 1 - 5.28T + 43T^{2} \) |
| 47 | \( 1 - 1.98T + 47T^{2} \) |
| 53 | \( 1 + 4.20T + 53T^{2} \) |
| 59 | \( 1 + 5.93T + 59T^{2} \) |
| 61 | \( 1 + 8.17T + 61T^{2} \) |
| 67 | \( 1 - 1.71T + 67T^{2} \) |
| 71 | \( 1 - 4.16T + 71T^{2} \) |
| 73 | \( 1 - 11.0T + 73T^{2} \) |
| 79 | \( 1 + 5.37T + 79T^{2} \) |
| 83 | \( 1 - 8.73T + 83T^{2} \) |
| 89 | \( 1 + 6.62T + 89T^{2} \) |
| 97 | \( 1 + 6.44T + 97T^{2} \) |
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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
−8.786241696535336336141720306224, −7.979381908804389114667043306076, −6.95993426294040768337231203837, −6.53199952324397662462457406105, −5.63300879994965409594685838400, −4.89916333012288435473050949059, −4.27113954283555978883193874720, −3.34558979732480648330080834084, −1.50302268950116795340134102381, −0.976268763879248203904474414928,
0.976268763879248203904474414928, 1.50302268950116795340134102381, 3.34558979732480648330080834084, 4.27113954283555978883193874720, 4.89916333012288435473050949059, 5.63300879994965409594685838400, 6.53199952324397662462457406105, 6.95993426294040768337231203837, 7.979381908804389114667043306076, 8.786241696535336336141720306224