L(s) = 1 | − 2.47·2-s + 3-s + 4.12·4-s + 0.0852·5-s − 2.47·6-s − 7-s − 5.26·8-s + 9-s − 0.211·10-s − 0.153·11-s + 4.12·12-s + 2.92·13-s + 2.47·14-s + 0.0852·15-s + 4.77·16-s − 3.51·17-s − 2.47·18-s − 2.81·19-s + 0.351·20-s − 21-s + 0.379·22-s + 4.67·23-s − 5.26·24-s − 4.99·25-s − 7.24·26-s + 27-s − 4.12·28-s + ⋯ |
L(s) = 1 | − 1.75·2-s + 0.577·3-s + 2.06·4-s + 0.0381·5-s − 1.01·6-s − 0.377·7-s − 1.86·8-s + 0.333·9-s − 0.0667·10-s − 0.0461·11-s + 1.19·12-s + 0.812·13-s + 0.661·14-s + 0.0220·15-s + 1.19·16-s − 0.853·17-s − 0.583·18-s − 0.645·19-s + 0.0787·20-s − 0.218·21-s + 0.0808·22-s + 0.975·23-s − 1.07·24-s − 0.998·25-s − 1.42·26-s + 0.192·27-s − 0.779·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7538011786\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7538011786\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 383 | \( 1 + T \) |
good | 2 | \( 1 + 2.47T + 2T^{2} \) |
| 5 | \( 1 - 0.0852T + 5T^{2} \) |
| 11 | \( 1 + 0.153T + 11T^{2} \) |
| 13 | \( 1 - 2.92T + 13T^{2} \) |
| 17 | \( 1 + 3.51T + 17T^{2} \) |
| 19 | \( 1 + 2.81T + 19T^{2} \) |
| 23 | \( 1 - 4.67T + 23T^{2} \) |
| 29 | \( 1 + 5.70T + 29T^{2} \) |
| 31 | \( 1 + 7.74T + 31T^{2} \) |
| 37 | \( 1 + 10.3T + 37T^{2} \) |
| 41 | \( 1 + 0.353T + 41T^{2} \) |
| 43 | \( 1 + 8.74T + 43T^{2} \) |
| 47 | \( 1 - 12.8T + 47T^{2} \) |
| 53 | \( 1 + 9.44T + 53T^{2} \) |
| 59 | \( 1 - 13.7T + 59T^{2} \) |
| 61 | \( 1 - 6.58T + 61T^{2} \) |
| 67 | \( 1 - 8.14T + 67T^{2} \) |
| 71 | \( 1 - 5.34T + 71T^{2} \) |
| 73 | \( 1 + 3.72T + 73T^{2} \) |
| 79 | \( 1 - 10.5T + 79T^{2} \) |
| 83 | \( 1 - 3.37T + 83T^{2} \) |
| 89 | \( 1 + 5.13T + 89T^{2} \) |
| 97 | \( 1 - 2.19T + 97T^{2} \) |
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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
−8.060468141529735128705343354794, −7.20769029832092343010293239447, −6.89368878904079621015457076591, −6.10451973758043902282999500391, −5.21176734542906760402943976492, −3.94060543765678563460772684064, −3.33471790207348577094666295023, −2.17708807823401312758418286886, −1.78460498874827495855630057226, −0.53001604535592692315325620526,
0.53001604535592692315325620526, 1.78460498874827495855630057226, 2.17708807823401312758418286886, 3.33471790207348577094666295023, 3.94060543765678563460772684064, 5.21176734542906760402943976492, 6.10451973758043902282999500391, 6.89368878904079621015457076591, 7.20769029832092343010293239447, 8.060468141529735128705343354794