L(s) = 1 | − 1.94·2-s + 1.97·3-s + 1.76·4-s − 3.82·6-s − 2.17·7-s + 0.458·8-s + 0.888·9-s − 4.36·11-s + 3.47·12-s − 1.66·13-s + 4.22·14-s − 4.41·16-s + 0.0716·17-s − 1.72·18-s + 1.98·19-s − 4.29·21-s + 8.46·22-s − 7.31·23-s + 0.903·24-s + 3.23·26-s − 4.16·27-s − 3.84·28-s + 4.53·29-s − 4.85·31-s + 7.65·32-s − 8.60·33-s − 0.138·34-s + ⋯ |
L(s) = 1 | − 1.37·2-s + 1.13·3-s + 0.881·4-s − 1.56·6-s − 0.822·7-s + 0.162·8-s + 0.296·9-s − 1.31·11-s + 1.00·12-s − 0.461·13-s + 1.12·14-s − 1.10·16-s + 0.0173·17-s − 0.406·18-s + 0.454·19-s − 0.937·21-s + 1.80·22-s − 1.52·23-s + 0.184·24-s + 0.633·26-s − 0.801·27-s − 0.725·28-s + 0.841·29-s − 0.871·31-s + 1.35·32-s − 1.49·33-s − 0.0238·34-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
\(L(1)\) |
\(\approx\) |
\(0.6803282243\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6803282243\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 1.94T + 2T^{2} \) |
| 3 | \( 1 - 1.97T + 3T^{2} \) |
| 7 | \( 1 + 2.17T + 7T^{2} \) |
| 11 | \( 1 + 4.36T + 11T^{2} \) |
| 13 | \( 1 + 1.66T + 13T^{2} \) |
| 17 | \( 1 - 0.0716T + 17T^{2} \) |
| 19 | \( 1 - 1.98T + 19T^{2} \) |
| 23 | \( 1 + 7.31T + 23T^{2} \) |
| 29 | \( 1 - 4.53T + 29T^{2} \) |
| 31 | \( 1 + 4.85T + 31T^{2} \) |
| 37 | \( 1 + 0.626T + 37T^{2} \) |
| 41 | \( 1 - 0.630T + 41T^{2} \) |
| 43 | \( 1 - 4.25T + 43T^{2} \) |
| 47 | \( 1 - 4.25T + 47T^{2} \) |
| 53 | \( 1 - 6.53T + 53T^{2} \) |
| 59 | \( 1 + 6.42T + 59T^{2} \) |
| 61 | \( 1 - 0.998T + 61T^{2} \) |
| 67 | \( 1 - 4.31T + 67T^{2} \) |
| 71 | \( 1 - 0.437T + 71T^{2} \) |
| 73 | \( 1 - 2.76T + 73T^{2} \) |
| 79 | \( 1 - 0.0176T + 79T^{2} \) |
| 83 | \( 1 - 7.97T + 83T^{2} \) |
| 89 | \( 1 + 3.58T + 89T^{2} \) |
| 97 | \( 1 - 9.52T + 97T^{2} \) |
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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
−8.123914401725509425323234888305, −7.68043462528986103603021384852, −7.14400820515010382261160667119, −6.15871052644324044466660779275, −5.29471043982232902401673414105, −4.24728645223151801193227722411, −3.30957025612493857567383739865, −2.54078279030608429598958065700, −1.95170607474472601424483297049, −0.47548983516840861612195329100,
0.47548983516840861612195329100, 1.95170607474472601424483297049, 2.54078279030608429598958065700, 3.30957025612493857567383739865, 4.24728645223151801193227722411, 5.29471043982232902401673414105, 6.15871052644324044466660779275, 7.14400820515010382261160667119, 7.68043462528986103603021384852, 8.123914401725509425323234888305