| L(s) = 1 | + 1.13·2-s + 3.01·3-s − 0.709·4-s − 5-s + 3.42·6-s + 3.60·7-s − 3.07·8-s + 6.11·9-s − 1.13·10-s + 11-s − 2.14·12-s − 2.72·13-s + 4.09·14-s − 3.01·15-s − 2.07·16-s + 6.65·17-s + 6.94·18-s + 3.17·19-s + 0.709·20-s + 10.8·21-s + 1.13·22-s − 3.83·23-s − 9.29·24-s + 25-s − 3.09·26-s + 9.38·27-s − 2.55·28-s + ⋯ |
| L(s) = 1 | + 0.803·2-s + 1.74·3-s − 0.354·4-s − 0.447·5-s + 1.39·6-s + 1.36·7-s − 1.08·8-s + 2.03·9-s − 0.359·10-s + 0.301·11-s − 0.618·12-s − 0.756·13-s + 1.09·14-s − 0.779·15-s − 0.519·16-s + 1.61·17-s + 1.63·18-s + 0.727·19-s + 0.158·20-s + 2.37·21-s + 0.242·22-s − 0.800·23-s − 1.89·24-s + 0.200·25-s − 0.607·26-s + 1.80·27-s − 0.483·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.290986841\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.290986841\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 73 | \( 1 + T \) |
| good | 2 | \( 1 - 1.13T + 2T^{2} \) |
| 3 | \( 1 - 3.01T + 3T^{2} \) |
| 7 | \( 1 - 3.60T + 7T^{2} \) |
| 13 | \( 1 + 2.72T + 13T^{2} \) |
| 17 | \( 1 - 6.65T + 17T^{2} \) |
| 19 | \( 1 - 3.17T + 19T^{2} \) |
| 23 | \( 1 + 3.83T + 23T^{2} \) |
| 29 | \( 1 - 1.05T + 29T^{2} \) |
| 31 | \( 1 - 6.95T + 31T^{2} \) |
| 37 | \( 1 + 3.71T + 37T^{2} \) |
| 41 | \( 1 + 0.117T + 41T^{2} \) |
| 43 | \( 1 + 5.22T + 43T^{2} \) |
| 47 | \( 1 - 0.328T + 47T^{2} \) |
| 53 | \( 1 - 0.867T + 53T^{2} \) |
| 59 | \( 1 + 5.51T + 59T^{2} \) |
| 61 | \( 1 + 5.58T + 61T^{2} \) |
| 67 | \( 1 - 1.81T + 67T^{2} \) |
| 71 | \( 1 - 3.78T + 71T^{2} \) |
| 79 | \( 1 + 6.00T + 79T^{2} \) |
| 83 | \( 1 - 0.912T + 83T^{2} \) |
| 89 | \( 1 - 13.3T + 89T^{2} \) |
| 97 | \( 1 - 10.7T + 97T^{2} \) |
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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
−8.417979517294269629158361676461, −7.74501748404950597921271802818, −7.42949138891644756680988051989, −6.12836335455607843074049562895, −4.99000318174390723932515737791, −4.65868219842184960826708269086, −3.68743177713055081079667010527, −3.23446335815141303028189694811, −2.27720083592216072937761276228, −1.19780273741564324503434512557,
1.19780273741564324503434512557, 2.27720083592216072937761276228, 3.23446335815141303028189694811, 3.68743177713055081079667010527, 4.65868219842184960826708269086, 4.99000318174390723932515737791, 6.12836335455607843074049562895, 7.42949138891644756680988051989, 7.74501748404950597921271802818, 8.417979517294269629158361676461
