| L(s) = 1 | − 2.77·2-s − 0.0161·3-s + 5.70·4-s + 0.0447·6-s − 4.43·7-s − 10.2·8-s − 2.99·9-s − 5.65·11-s − 0.0920·12-s + 0.883·13-s + 12.3·14-s + 17.1·16-s + 4.14·17-s + 8.32·18-s + 0.923·19-s + 0.0715·21-s + 15.7·22-s + 4.69·23-s + 0.165·24-s − 2.45·26-s + 0.0967·27-s − 25.3·28-s + 2.90·29-s − 8.10·31-s − 27.0·32-s + 0.0912·33-s − 11.5·34-s + ⋯ |
| L(s) = 1 | − 1.96·2-s − 0.00931·3-s + 2.85·4-s + 0.0182·6-s − 1.67·7-s − 3.63·8-s − 0.999·9-s − 1.70·11-s − 0.0265·12-s + 0.245·13-s + 3.29·14-s + 4.28·16-s + 1.00·17-s + 1.96·18-s + 0.211·19-s + 0.0156·21-s + 3.34·22-s + 0.978·23-s + 0.0338·24-s − 0.481·26-s + 0.0186·27-s − 4.78·28-s + 0.538·29-s − 1.45·31-s − 4.77·32-s + 0.0158·33-s − 1.97·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 241 | \( 1 + T \) |
| good | 2 | \( 1 + 2.77T + 2T^{2} \) |
| 3 | \( 1 + 0.0161T + 3T^{2} \) |
| 7 | \( 1 + 4.43T + 7T^{2} \) |
| 11 | \( 1 + 5.65T + 11T^{2} \) |
| 13 | \( 1 - 0.883T + 13T^{2} \) |
| 17 | \( 1 - 4.14T + 17T^{2} \) |
| 19 | \( 1 - 0.923T + 19T^{2} \) |
| 23 | \( 1 - 4.69T + 23T^{2} \) |
| 29 | \( 1 - 2.90T + 29T^{2} \) |
| 31 | \( 1 + 8.10T + 31T^{2} \) |
| 37 | \( 1 - 10.7T + 37T^{2} \) |
| 41 | \( 1 + 0.571T + 41T^{2} \) |
| 43 | \( 1 + 0.399T + 43T^{2} \) |
| 47 | \( 1 + 6.27T + 47T^{2} \) |
| 53 | \( 1 + 3.12T + 53T^{2} \) |
| 59 | \( 1 + 0.915T + 59T^{2} \) |
| 61 | \( 1 - 9.75T + 61T^{2} \) |
| 67 | \( 1 + 9.46T + 67T^{2} \) |
| 71 | \( 1 + 0.00570T + 71T^{2} \) |
| 73 | \( 1 + 0.642T + 73T^{2} \) |
| 79 | \( 1 - 5.67T + 79T^{2} \) |
| 83 | \( 1 + 7.08T + 83T^{2} \) |
| 89 | \( 1 - 0.805T + 89T^{2} \) |
| 97 | \( 1 - 17.3T + 97T^{2} \) |
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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
−7.76590055827929534652615549424, −7.36660061859819970075771424097, −6.47694214424839899297627914940, −5.92081393655633852554714785821, −5.30170703128649918103071659724, −3.28615131396741496092015668645, −3.03792089812920977431310585787, −2.27006882684991019061060316947, −0.816420907164703419535043226922, 0,
0.816420907164703419535043226922, 2.27006882684991019061060316947, 3.03792089812920977431310585787, 3.28615131396741496092015668645, 5.30170703128649918103071659724, 5.92081393655633852554714785821, 6.47694214424839899297627914940, 7.36660061859819970075771424097, 7.76590055827929534652615549424
