L(s) = 1 | − 2.23·2-s + 3-s + 3.00·4-s − 2.23·6-s − 2.23·7-s − 2.23·8-s + 9-s + 3.00·12-s − 4.47·13-s + 5.00·14-s − 0.999·16-s − 2.23·17-s − 2.23·18-s − 2.23·19-s − 2.23·21-s + 23-s − 2.23·24-s + 10.0·26-s + 27-s − 6.70·28-s + 4.47·29-s + 10·31-s + 6.70·32-s + 5.00·34-s + 3.00·36-s + 7·37-s + 5.00·38-s + ⋯ |
L(s) = 1 | − 1.58·2-s + 0.577·3-s + 1.50·4-s − 0.912·6-s − 0.845·7-s − 0.790·8-s + 0.333·9-s + 0.866·12-s − 1.24·13-s + 1.33·14-s − 0.249·16-s − 0.542·17-s − 0.527·18-s − 0.512·19-s − 0.487·21-s + 0.208·23-s − 0.456·24-s + 1.96·26-s + 0.192·27-s − 1.26·28-s + 0.830·29-s + 1.79·31-s + 1.18·32-s + 0.857·34-s + 0.500·36-s + 1.15·37-s + 0.811·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9075 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9075 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6697670900\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6697670900\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + 2.23T + 2T^{2} \) |
| 7 | \( 1 + 2.23T + 7T^{2} \) |
| 13 | \( 1 + 4.47T + 13T^{2} \) |
| 17 | \( 1 + 2.23T + 17T^{2} \) |
| 19 | \( 1 + 2.23T + 19T^{2} \) |
| 23 | \( 1 - T + 23T^{2} \) |
| 29 | \( 1 - 4.47T + 29T^{2} \) |
| 31 | \( 1 - 10T + 31T^{2} \) |
| 37 | \( 1 - 7T + 37T^{2} \) |
| 41 | \( 1 - 6.70T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 - 3T + 47T^{2} \) |
| 53 | \( 1 + 14T + 53T^{2} \) |
| 59 | \( 1 + 5T + 59T^{2} \) |
| 61 | \( 1 - 4.47T + 61T^{2} \) |
| 67 | \( 1 + 2T + 67T^{2} \) |
| 71 | \( 1 + 13T + 71T^{2} \) |
| 73 | \( 1 + 13.4T + 73T^{2} \) |
| 79 | \( 1 + 15.6T + 79T^{2} \) |
| 83 | \( 1 + 4.47T + 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 - 17T + 97T^{2} \) |
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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
−7.67177847332852898395802214989, −7.47599123188568249353564329101, −6.48123883854689824780379347070, −6.23396478766626397164676707107, −4.76246636135586909367113100201, −4.28362891657510522251442451003, −2.81971551900331192733879873918, −2.68699381446426908947051633096, −1.56605571136561244184121519252, −0.48796540252597017709798139400,
0.48796540252597017709798139400, 1.56605571136561244184121519252, 2.68699381446426908947051633096, 2.81971551900331192733879873918, 4.28362891657510522251442451003, 4.76246636135586909367113100201, 6.23396478766626397164676707107, 6.48123883854689824780379347070, 7.47599123188568249353564329101, 7.67177847332852898395802214989