| L(s) = 1 | + 2.37·2-s − 1.58·3-s + 3.63·4-s + 5-s − 3.76·6-s − 7-s + 3.86·8-s − 0.480·9-s + 2.37·10-s + 2.98·11-s − 5.76·12-s + 3.38·13-s − 2.37·14-s − 1.58·15-s + 1.91·16-s − 1.78·17-s − 1.14·18-s − 7.78·19-s + 3.63·20-s + 1.58·21-s + 7.08·22-s − 2.47·23-s − 6.13·24-s + 25-s + 8.03·26-s + 5.52·27-s − 3.63·28-s + ⋯ |
| L(s) = 1 | + 1.67·2-s − 0.916·3-s + 1.81·4-s + 0.447·5-s − 1.53·6-s − 0.377·7-s + 1.36·8-s − 0.160·9-s + 0.750·10-s + 0.900·11-s − 1.66·12-s + 0.939·13-s − 0.634·14-s − 0.409·15-s + 0.479·16-s − 0.433·17-s − 0.268·18-s − 1.78·19-s + 0.811·20-s + 0.346·21-s + 1.51·22-s − 0.517·23-s − 1.25·24-s + 0.200·25-s + 1.57·26-s + 1.06·27-s − 0.686·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 229 | \( 1 - T \) |
| good | 2 | \( 1 - 2.37T + 2T^{2} \) |
| 3 | \( 1 + 1.58T + 3T^{2} \) |
| 11 | \( 1 - 2.98T + 11T^{2} \) |
| 13 | \( 1 - 3.38T + 13T^{2} \) |
| 17 | \( 1 + 1.78T + 17T^{2} \) |
| 19 | \( 1 + 7.78T + 19T^{2} \) |
| 23 | \( 1 + 2.47T + 23T^{2} \) |
| 29 | \( 1 + 1.68T + 29T^{2} \) |
| 31 | \( 1 + 9.55T + 31T^{2} \) |
| 37 | \( 1 + 4.50T + 37T^{2} \) |
| 41 | \( 1 - 9.20T + 41T^{2} \) |
| 43 | \( 1 + 4.27T + 43T^{2} \) |
| 47 | \( 1 + 11.1T + 47T^{2} \) |
| 53 | \( 1 + 6.82T + 53T^{2} \) |
| 59 | \( 1 + 1.17T + 59T^{2} \) |
| 61 | \( 1 + 7.35T + 61T^{2} \) |
| 67 | \( 1 - 10.3T + 67T^{2} \) |
| 71 | \( 1 - 15.8T + 71T^{2} \) |
| 73 | \( 1 + 3.82T + 73T^{2} \) |
| 79 | \( 1 - 13.6T + 79T^{2} \) |
| 83 | \( 1 - 10.2T + 83T^{2} \) |
| 89 | \( 1 + 8.64T + 89T^{2} \) |
| 97 | \( 1 + 9.04T + 97T^{2} \) |
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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
−6.76484777130212152714243505458, −6.49983405439710517990361587500, −6.07467907890913524219798844462, −5.42225644320115065461232019635, −4.75443056926753202245128984696, −3.93158911116221528447633358735, −3.49749616289183708366368685029, −2.37391471805808960569171555126, −1.60541544178015992880436523862, 0,
1.60541544178015992880436523862, 2.37391471805808960569171555126, 3.49749616289183708366368685029, 3.93158911116221528447633358735, 4.75443056926753202245128984696, 5.42225644320115065461232019635, 6.07467907890913524219798844462, 6.49983405439710517990361587500, 6.76484777130212152714243505458
