| L(s) = 1 | − 2-s − 4-s + 4.47·7-s + 3·8-s − 3.23·11-s + 3.38·13-s − 4.47·14-s − 16-s + 2.85·17-s − 3.23·19-s + 3.23·22-s + 4.47·23-s − 3.38·26-s − 4.47·28-s − 4.38·29-s + 7.23·31-s − 5·32-s − 2.85·34-s + 8.09·37-s + 3.23·38-s + 1.38·41-s − 5.70·43-s + 3.23·44-s − 4.47·46-s + 5.23·47-s + 13.0·49-s − 3.38·52-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.5·4-s + 1.69·7-s + 1.06·8-s − 0.975·11-s + 0.937·13-s − 1.19·14-s − 0.250·16-s + 0.692·17-s − 0.742·19-s + 0.689·22-s + 0.932·23-s − 0.663·26-s − 0.845·28-s − 0.813·29-s + 1.29·31-s − 0.883·32-s − 0.489·34-s + 1.33·37-s + 0.524·38-s + 0.215·41-s − 0.870·43-s + 0.487·44-s − 0.659·46-s + 0.763·47-s + 1.85·49-s − 0.468·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.510305442\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.510305442\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + T + 2T^{2} \) |
| 7 | \( 1 - 4.47T + 7T^{2} \) |
| 11 | \( 1 + 3.23T + 11T^{2} \) |
| 13 | \( 1 - 3.38T + 13T^{2} \) |
| 17 | \( 1 - 2.85T + 17T^{2} \) |
| 19 | \( 1 + 3.23T + 19T^{2} \) |
| 23 | \( 1 - 4.47T + 23T^{2} \) |
| 29 | \( 1 + 4.38T + 29T^{2} \) |
| 31 | \( 1 - 7.23T + 31T^{2} \) |
| 37 | \( 1 - 8.09T + 37T^{2} \) |
| 41 | \( 1 - 1.38T + 41T^{2} \) |
| 43 | \( 1 + 5.70T + 43T^{2} \) |
| 47 | \( 1 - 5.23T + 47T^{2} \) |
| 53 | \( 1 + 1.38T + 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 + 0.618T + 61T^{2} \) |
| 67 | \( 1 - 5.23T + 67T^{2} \) |
| 71 | \( 1 - 0.763T + 71T^{2} \) |
| 73 | \( 1 - 3.09T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 3.52T + 83T^{2} \) |
| 89 | \( 1 + 7.61T + 89T^{2} \) |
| 97 | \( 1 - 8.85T + 97T^{2} \) |
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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
−8.068879158137820640600228864461, −7.889608784936010687153318818565, −6.98220643525069411866824949071, −5.84573387921575067050587818701, −5.15415647227732069674956261999, −4.58895021170904813033880785010, −3.84105478798141704607848685073, −2.58856050799556273377415841197, −1.57278414419040865392828956051, −0.804413271596880927503481744996,
0.804413271596880927503481744996, 1.57278414419040865392828956051, 2.58856050799556273377415841197, 3.84105478798141704607848685073, 4.58895021170904813033880785010, 5.15415647227732069674956261999, 5.84573387921575067050587818701, 6.98220643525069411866824949071, 7.889608784936010687153318818565, 8.068879158137820640600228864461
