L(s) = 1 | + 2-s − 0.370·3-s + 4-s + 2.36·5-s − 0.370·6-s + 1.74·7-s + 8-s − 2.86·9-s + 2.36·10-s − 0.565·11-s − 0.370·12-s − 1.80·13-s + 1.74·14-s − 0.876·15-s + 16-s + 4.37·17-s − 2.86·18-s + 2.88·19-s + 2.36·20-s − 0.646·21-s − 0.565·22-s − 0.747·23-s − 0.370·24-s + 0.596·25-s − 1.80·26-s + 2.17·27-s + 1.74·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.213·3-s + 0.5·4-s + 1.05·5-s − 0.151·6-s + 0.660·7-s + 0.353·8-s − 0.954·9-s + 0.748·10-s − 0.170·11-s − 0.106·12-s − 0.500·13-s + 0.466·14-s − 0.226·15-s + 0.250·16-s + 1.06·17-s − 0.674·18-s + 0.661·19-s + 0.528·20-s − 0.141·21-s − 0.120·22-s − 0.155·23-s − 0.0756·24-s + 0.119·25-s − 0.353·26-s + 0.417·27-s + 0.330·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.887722335\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.887722335\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 31 | \( 1 + T \) |
| 97 | \( 1 + T \) |
good | 3 | \( 1 + 0.370T + 3T^{2} \) |
| 5 | \( 1 - 2.36T + 5T^{2} \) |
| 7 | \( 1 - 1.74T + 7T^{2} \) |
| 11 | \( 1 + 0.565T + 11T^{2} \) |
| 13 | \( 1 + 1.80T + 13T^{2} \) |
| 17 | \( 1 - 4.37T + 17T^{2} \) |
| 19 | \( 1 - 2.88T + 19T^{2} \) |
| 23 | \( 1 + 0.747T + 23T^{2} \) |
| 29 | \( 1 - 7.97T + 29T^{2} \) |
| 37 | \( 1 - 4.19T + 37T^{2} \) |
| 41 | \( 1 + 3.11T + 41T^{2} \) |
| 43 | \( 1 - 3.55T + 43T^{2} \) |
| 47 | \( 1 + 0.928T + 47T^{2} \) |
| 53 | \( 1 + 5.73T + 53T^{2} \) |
| 59 | \( 1 - 9.35T + 59T^{2} \) |
| 61 | \( 1 - 4.49T + 61T^{2} \) |
| 67 | \( 1 + 3.08T + 67T^{2} \) |
| 71 | \( 1 - 10.3T + 71T^{2} \) |
| 73 | \( 1 - 0.723T + 73T^{2} \) |
| 79 | \( 1 + 3.87T + 79T^{2} \) |
| 83 | \( 1 + 1.70T + 83T^{2} \) |
| 89 | \( 1 - 12.3T + 89T^{2} \) |
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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
−8.004219464614105438358387289228, −7.32355230600747057585923744440, −6.33255512117815800620104847958, −5.88666820147865624185436124614, −5.14864768294367464109518918109, −4.81491563210180056183217102853, −3.56306916730979067326333650756, −2.76520040827713227347210298771, −2.04994076466606593587258674280, −0.971099858912182739183554789284,
0.971099858912182739183554789284, 2.04994076466606593587258674280, 2.76520040827713227347210298771, 3.56306916730979067326333650756, 4.81491563210180056183217102853, 5.14864768294367464109518918109, 5.88666820147865624185436124614, 6.33255512117815800620104847958, 7.32355230600747057585923744440, 8.004219464614105438358387289228