| L(s) = 1 | − 0.751·3-s − 0.661·7-s − 2.43·9-s + 5.38·11-s + 3.45·13-s + 6.04·17-s + 1.38·19-s + 0.497·21-s + 23-s + 4.08·27-s − 3.34·29-s + 0.863·31-s − 4.04·33-s + 3.82·37-s − 2.59·39-s − 1.91·41-s − 5.88·43-s − 4.11·47-s − 6.56·49-s − 4.54·51-s + 3.00·53-s − 1.03·57-s − 14.1·59-s − 8.26·61-s + 1.61·63-s − 2.49·67-s − 0.751·69-s + ⋯ |
| L(s) = 1 | − 0.433·3-s − 0.250·7-s − 0.811·9-s + 1.62·11-s + 0.956·13-s + 1.46·17-s + 0.317·19-s + 0.108·21-s + 0.208·23-s + 0.785·27-s − 0.621·29-s + 0.155·31-s − 0.704·33-s + 0.628·37-s − 0.415·39-s − 0.298·41-s − 0.896·43-s − 0.599·47-s − 0.937·49-s − 0.635·51-s + 0.412·53-s − 0.137·57-s − 1.83·59-s − 1.05·61-s + 0.203·63-s − 0.305·67-s − 0.0904·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.799980189\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.799980189\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 + 0.751T + 3T^{2} \) |
| 7 | \( 1 + 0.661T + 7T^{2} \) |
| 11 | \( 1 - 5.38T + 11T^{2} \) |
| 13 | \( 1 - 3.45T + 13T^{2} \) |
| 17 | \( 1 - 6.04T + 17T^{2} \) |
| 19 | \( 1 - 1.38T + 19T^{2} \) |
| 29 | \( 1 + 3.34T + 29T^{2} \) |
| 31 | \( 1 - 0.863T + 31T^{2} \) |
| 37 | \( 1 - 3.82T + 37T^{2} \) |
| 41 | \( 1 + 1.91T + 41T^{2} \) |
| 43 | \( 1 + 5.88T + 43T^{2} \) |
| 47 | \( 1 + 4.11T + 47T^{2} \) |
| 53 | \( 1 - 3.00T + 53T^{2} \) |
| 59 | \( 1 + 14.1T + 59T^{2} \) |
| 61 | \( 1 + 8.26T + 61T^{2} \) |
| 67 | \( 1 + 2.49T + 67T^{2} \) |
| 71 | \( 1 - 6.65T + 71T^{2} \) |
| 73 | \( 1 - 12.7T + 73T^{2} \) |
| 79 | \( 1 - 8.64T + 79T^{2} \) |
| 83 | \( 1 - 5.06T + 83T^{2} \) |
| 89 | \( 1 - 2.22T + 89T^{2} \) |
| 97 | \( 1 - 14.7T + 97T^{2} \) |
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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
−8.300874392830742238048283935903, −7.66366637097210653845734502689, −6.57521283233098513327925663234, −6.23464910480669337845059149658, −5.51459023768900616127318434825, −4.66318341097529455138970890533, −3.53956854095953229799701582006, −3.23864966623638943474634277826, −1.70108657242288926879417531633, −0.809473635886224542881266383255,
0.809473635886224542881266383255, 1.70108657242288926879417531633, 3.23864966623638943474634277826, 3.53956854095953229799701582006, 4.66318341097529455138970890533, 5.51459023768900616127318434825, 6.23464910480669337845059149658, 6.57521283233098513327925663234, 7.66366637097210653845734502689, 8.300874392830742238048283935903
