L(s) = 1 | − 1.45·3-s + 0.900·5-s − 0.890·9-s + 4.36·11-s + 1.64·13-s − 1.30·15-s + 3.44·17-s + 5.80·19-s − 3.72·23-s − 4.18·25-s + 5.65·27-s − 1.84·29-s − 4.01·31-s − 6.33·33-s − 9.44·37-s − 2.38·39-s + 9.12·41-s + 6.15·43-s − 0.801·45-s − 5.94·47-s − 5.00·51-s + 2.51·53-s + 3.92·55-s − 8.43·57-s + 8.10·59-s − 6.73·61-s + 1.47·65-s + ⋯ |
L(s) = 1 | − 0.838·3-s + 0.402·5-s − 0.296·9-s + 1.31·11-s + 0.455·13-s − 0.337·15-s + 0.836·17-s + 1.33·19-s − 0.775·23-s − 0.837·25-s + 1.08·27-s − 0.342·29-s − 0.720·31-s − 1.10·33-s − 1.55·37-s − 0.381·39-s + 1.42·41-s + 0.938·43-s − 0.119·45-s − 0.866·47-s − 0.701·51-s + 0.345·53-s + 0.529·55-s − 1.11·57-s + 1.05·59-s − 0.861·61-s + 0.183·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2744 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.538708647\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.538708647\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + 1.45T + 3T^{2} \) |
| 5 | \( 1 - 0.900T + 5T^{2} \) |
| 11 | \( 1 - 4.36T + 11T^{2} \) |
| 13 | \( 1 - 1.64T + 13T^{2} \) |
| 17 | \( 1 - 3.44T + 17T^{2} \) |
| 19 | \( 1 - 5.80T + 19T^{2} \) |
| 23 | \( 1 + 3.72T + 23T^{2} \) |
| 29 | \( 1 + 1.84T + 29T^{2} \) |
| 31 | \( 1 + 4.01T + 31T^{2} \) |
| 37 | \( 1 + 9.44T + 37T^{2} \) |
| 41 | \( 1 - 9.12T + 41T^{2} \) |
| 43 | \( 1 - 6.15T + 43T^{2} \) |
| 47 | \( 1 + 5.94T + 47T^{2} \) |
| 53 | \( 1 - 2.51T + 53T^{2} \) |
| 59 | \( 1 - 8.10T + 59T^{2} \) |
| 61 | \( 1 + 6.73T + 61T^{2} \) |
| 67 | \( 1 - 9.52T + 67T^{2} \) |
| 71 | \( 1 - 2.48T + 71T^{2} \) |
| 73 | \( 1 + 6.74T + 73T^{2} \) |
| 79 | \( 1 - 8.72T + 79T^{2} \) |
| 83 | \( 1 - 6.65T + 83T^{2} \) |
| 89 | \( 1 + 3.00T + 89T^{2} \) |
| 97 | \( 1 - 17.1T + 97T^{2} \) |
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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
−9.005690618196821598554561456300, −8.011022135149421442812570821156, −7.20574865464792022622495577349, −6.32558456556589523314454561967, −5.76063205523098855341732964785, −5.21230378145680076367374929087, −4.00566902615856114289363585325, −3.29620803716977305461372477384, −1.88007196685903991755470137623, −0.837655489942793357065241555824,
0.837655489942793357065241555824, 1.88007196685903991755470137623, 3.29620803716977305461372477384, 4.00566902615856114289363585325, 5.21230378145680076367374929087, 5.76063205523098855341732964785, 6.32558456556589523314454561967, 7.20574865464792022622495577349, 8.011022135149421442812570821156, 9.005690618196821598554561456300