| L(s) = 1 | − 2.15·3-s + 1.57·5-s + 2.46·7-s + 1.64·9-s − 5.90·11-s − 3.94·13-s − 3.40·15-s + 1.56·19-s − 5.31·21-s − 1.12·23-s − 2.51·25-s + 2.91·27-s − 8.23·29-s + 4.16·31-s + 12.7·33-s + 3.88·35-s − 8.28·37-s + 8.49·39-s − 8.29·41-s + 4.88·43-s + 2.60·45-s + 0.211·47-s − 0.921·49-s − 2.51·53-s − 9.31·55-s − 3.37·57-s − 6.97·59-s + ⋯ |
| L(s) = 1 | − 1.24·3-s + 0.705·5-s + 0.931·7-s + 0.549·9-s − 1.77·11-s − 1.09·13-s − 0.878·15-s + 0.359·19-s − 1.16·21-s − 0.235·23-s − 0.502·25-s + 0.560·27-s − 1.53·29-s + 0.748·31-s + 2.21·33-s + 0.657·35-s − 1.36·37-s + 1.36·39-s − 1.29·41-s + 0.745·43-s + 0.387·45-s + 0.0308·47-s − 0.131·49-s − 0.345·53-s − 1.25·55-s − 0.447·57-s − 0.907·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9248 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9248 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7710494213\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7710494213\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 \) |
| good | 3 | \( 1 + 2.15T + 3T^{2} \) |
| 5 | \( 1 - 1.57T + 5T^{2} \) |
| 7 | \( 1 - 2.46T + 7T^{2} \) |
| 11 | \( 1 + 5.90T + 11T^{2} \) |
| 13 | \( 1 + 3.94T + 13T^{2} \) |
| 19 | \( 1 - 1.56T + 19T^{2} \) |
| 23 | \( 1 + 1.12T + 23T^{2} \) |
| 29 | \( 1 + 8.23T + 29T^{2} \) |
| 31 | \( 1 - 4.16T + 31T^{2} \) |
| 37 | \( 1 + 8.28T + 37T^{2} \) |
| 41 | \( 1 + 8.29T + 41T^{2} \) |
| 43 | \( 1 - 4.88T + 43T^{2} \) |
| 47 | \( 1 - 0.211T + 47T^{2} \) |
| 53 | \( 1 + 2.51T + 53T^{2} \) |
| 59 | \( 1 + 6.97T + 59T^{2} \) |
| 61 | \( 1 + 0.958T + 61T^{2} \) |
| 67 | \( 1 + 2.23T + 67T^{2} \) |
| 71 | \( 1 - 10.3T + 71T^{2} \) |
| 73 | \( 1 - 8.62T + 73T^{2} \) |
| 79 | \( 1 - 11.2T + 79T^{2} \) |
| 83 | \( 1 - 13.3T + 83T^{2} \) |
| 89 | \( 1 - 9.61T + 89T^{2} \) |
| 97 | \( 1 + 10.2T + 97T^{2} \) |
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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
−7.74184814206746262810931564981, −7.00007433178702523199219047972, −6.15252731658112768763523516463, −5.49680821807790129168861199876, −5.06169326896119219312810983221, −4.75373265284189903016073263177, −3.42552777550793980609184964582, −2.35738066051313473267115270530, −1.79394824065389871189177169715, −0.43209502573715355666266776088,
0.43209502573715355666266776088, 1.79394824065389871189177169715, 2.35738066051313473267115270530, 3.42552777550793980609184964582, 4.75373265284189903016073263177, 5.06169326896119219312810983221, 5.49680821807790129168861199876, 6.15252731658112768763523516463, 7.00007433178702523199219047972, 7.74184814206746262810931564981
