L(s) = 1 | − 1.94·2-s + 1.79·4-s − 3.28·5-s − 0.265·7-s + 0.393·8-s + 6.39·10-s − 4.32·11-s + 5.51·13-s + 0.517·14-s − 4.36·16-s − 2.77·17-s + 19-s − 5.90·20-s + 8.42·22-s − 7.94·23-s + 5.77·25-s − 10.7·26-s − 0.477·28-s + 8.37·29-s − 7.81·31-s + 7.71·32-s + 5.40·34-s + 0.870·35-s + 4.10·37-s − 1.94·38-s − 1.29·40-s + 0.744·41-s + ⋯ |
L(s) = 1 | − 1.37·2-s + 0.899·4-s − 1.46·5-s − 0.100·7-s + 0.139·8-s + 2.02·10-s − 1.30·11-s + 1.52·13-s + 0.138·14-s − 1.09·16-s − 0.672·17-s + 0.229·19-s − 1.31·20-s + 1.79·22-s − 1.65·23-s + 1.15·25-s − 2.10·26-s − 0.0901·28-s + 1.55·29-s − 1.40·31-s + 1.36·32-s + 0.926·34-s + 0.147·35-s + 0.675·37-s − 0.316·38-s − 0.204·40-s + 0.116·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8037 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8037 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 19 | \( 1 - T \) |
| 47 | \( 1 - T \) |
good | 2 | \( 1 + 1.94T + 2T^{2} \) |
| 5 | \( 1 + 3.28T + 5T^{2} \) |
| 7 | \( 1 + 0.265T + 7T^{2} \) |
| 11 | \( 1 + 4.32T + 11T^{2} \) |
| 13 | \( 1 - 5.51T + 13T^{2} \) |
| 17 | \( 1 + 2.77T + 17T^{2} \) |
| 23 | \( 1 + 7.94T + 23T^{2} \) |
| 29 | \( 1 - 8.37T + 29T^{2} \) |
| 31 | \( 1 + 7.81T + 31T^{2} \) |
| 37 | \( 1 - 4.10T + 37T^{2} \) |
| 41 | \( 1 - 0.744T + 41T^{2} \) |
| 43 | \( 1 - 6.11T + 43T^{2} \) |
| 53 | \( 1 - 6.56T + 53T^{2} \) |
| 59 | \( 1 + 1.64T + 59T^{2} \) |
| 61 | \( 1 - 4.87T + 61T^{2} \) |
| 67 | \( 1 + 3.90T + 67T^{2} \) |
| 71 | \( 1 - 0.591T + 71T^{2} \) |
| 73 | \( 1 - 3.33T + 73T^{2} \) |
| 79 | \( 1 - 0.757T + 79T^{2} \) |
| 83 | \( 1 + 0.0612T + 83T^{2} \) |
| 89 | \( 1 + 2.87T + 89T^{2} \) |
| 97 | \( 1 - 3.42T + 97T^{2} \) |
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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
−7.87355639384945973894671608719, −7.12625299104494388502932027282, −6.38527579526895259780636214969, −5.49908053877363045046882166619, −4.41827188561408619055093461690, −3.95826450745616701323315750092, −2.98806647975014238545495192875, −1.99884367413888841154206718504, −0.828001581502976169238383937483, 0,
0.828001581502976169238383937483, 1.99884367413888841154206718504, 2.98806647975014238545495192875, 3.95826450745616701323315750092, 4.41827188561408619055093461690, 5.49908053877363045046882166619, 6.38527579526895259780636214969, 7.12625299104494388502932027282, 7.87355639384945973894671608719