L(s) = 1 | + 2.49·2-s + 3-s + 4.23·4-s + 2.49·6-s + 5.58·8-s + 9-s − 4.47·11-s + 4.23·12-s + 5.23·13-s + 5.47·16-s + 0.763·17-s + 2.49·18-s + 8.08·19-s − 11.1·22-s + 3.08·23-s + 5.58·24-s + 13.0·26-s + 27-s + 2·29-s − 1.90·31-s + 2.49·32-s − 4.47·33-s + 1.90·34-s + 4.23·36-s − 6.17·37-s + 20.1·38-s + 5.23·39-s + ⋯ |
L(s) = 1 | + 1.76·2-s + 0.577·3-s + 2.11·4-s + 1.01·6-s + 1.97·8-s + 0.333·9-s − 1.34·11-s + 1.22·12-s + 1.45·13-s + 1.36·16-s + 0.185·17-s + 0.588·18-s + 1.85·19-s − 2.38·22-s + 0.643·23-s + 1.13·24-s + 2.56·26-s + 0.192·27-s + 0.371·29-s − 0.342·31-s + 0.441·32-s − 0.778·33-s + 0.327·34-s + 0.706·36-s − 1.01·37-s + 3.27·38-s + 0.838·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(7.380813026\) |
\(L(\frac12)\) |
\(\approx\) |
\(7.380813026\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 - 2.49T + 2T^{2} \) |
| 11 | \( 1 + 4.47T + 11T^{2} \) |
| 13 | \( 1 - 5.23T + 13T^{2} \) |
| 17 | \( 1 - 0.763T + 17T^{2} \) |
| 19 | \( 1 - 8.08T + 19T^{2} \) |
| 23 | \( 1 - 3.08T + 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 + 1.90T + 31T^{2} \) |
| 37 | \( 1 + 6.17T + 37T^{2} \) |
| 41 | \( 1 - 6.90T + 41T^{2} \) |
| 43 | \( 1 + 9.98T + 43T^{2} \) |
| 47 | \( 1 + 4.94T + 47T^{2} \) |
| 53 | \( 1 - 1.90T + 53T^{2} \) |
| 59 | \( 1 + 9.98T + 59T^{2} \) |
| 61 | \( 1 - 3.81T + 61T^{2} \) |
| 67 | \( 1 - 6.17T + 67T^{2} \) |
| 71 | \( 1 - 0.472T + 71T^{2} \) |
| 73 | \( 1 - 2.76T + 73T^{2} \) |
| 79 | \( 1 - 12.9T + 79T^{2} \) |
| 83 | \( 1 - 6.47T + 83T^{2} \) |
| 89 | \( 1 + 9.26T + 89T^{2} \) |
| 97 | \( 1 - 10.1T + 97T^{2} \) |
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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
−8.277100846146341390705274234476, −7.62288927702962663358223945903, −6.89912881377829920988484774861, −6.07840509202526184102413631446, −5.27019487781283009607866357951, −4.88914286833448265090545321858, −3.66045409778749543469123830850, −3.30126774344454311097563817123, −2.50263008206974355980968714904, −1.35054764093303449612626188804,
1.35054764093303449612626188804, 2.50263008206974355980968714904, 3.30126774344454311097563817123, 3.66045409778749543469123830850, 4.88914286833448265090545321858, 5.27019487781283009607866357951, 6.07840509202526184102413631446, 6.89912881377829920988484774861, 7.62288927702962663358223945903, 8.277100846146341390705274234476