L(s) = 1 | − 2-s − 3-s + 4-s − 0.307·5-s + 6-s − 8-s + 9-s + 0.307·10-s + 1.25·11-s − 12-s + 13-s + 0.307·15-s + 16-s + 2.64·17-s − 18-s + 7.86·19-s − 0.307·20-s − 1.25·22-s − 6.47·23-s + 24-s − 4.90·25-s − 26-s − 27-s + 8.88·29-s − 0.307·30-s − 5.22·31-s − 32-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.137·5-s + 0.408·6-s − 0.353·8-s + 0.333·9-s + 0.0973·10-s + 0.379·11-s − 0.288·12-s + 0.277·13-s + 0.0794·15-s + 0.250·16-s + 0.640·17-s − 0.235·18-s + 1.80·19-s − 0.0688·20-s − 0.268·22-s − 1.35·23-s + 0.204·24-s − 0.981·25-s − 0.196·26-s − 0.192·27-s + 1.64·29-s − 0.0561·30-s − 0.937·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3822 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3822 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.107207912\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.107207912\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 13 | \( 1 - T \) |
good | 5 | \( 1 + 0.307T + 5T^{2} \) |
| 11 | \( 1 - 1.25T + 11T^{2} \) |
| 17 | \( 1 - 2.64T + 17T^{2} \) |
| 19 | \( 1 - 7.86T + 19T^{2} \) |
| 23 | \( 1 + 6.47T + 23T^{2} \) |
| 29 | \( 1 - 8.88T + 29T^{2} \) |
| 31 | \( 1 + 5.22T + 31T^{2} \) |
| 37 | \( 1 - 8.44T + 37T^{2} \) |
| 41 | \( 1 + 0.192T + 41T^{2} \) |
| 43 | \( 1 - 10.4T + 43T^{2} \) |
| 47 | \( 1 + 9.74T + 47T^{2} \) |
| 53 | \( 1 - 7.74T + 53T^{2} \) |
| 59 | \( 1 + 10.0T + 59T^{2} \) |
| 61 | \( 1 + 9.38T + 61T^{2} \) |
| 67 | \( 1 - 9.62T + 67T^{2} \) |
| 71 | \( 1 + 1.56T + 71T^{2} \) |
| 73 | \( 1 + 6.83T + 73T^{2} \) |
| 79 | \( 1 + 16.0T + 79T^{2} \) |
| 83 | \( 1 - 1.04T + 83T^{2} \) |
| 89 | \( 1 + 1.24T + 89T^{2} \) |
| 97 | \( 1 - 15.7T + 97T^{2} \) |
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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
−8.429873255960323474871716302272, −7.67106396526759761595204212560, −7.27506634509109281592273390108, −6.09941866119697653779583060823, −5.86637308739081627632196487749, −4.75959099094212607849065127461, −3.82813941873210653378196391269, −2.91893768888198608697520250989, −1.65801602236660685962059142644, −0.73173820994508540256734422414,
0.73173820994508540256734422414, 1.65801602236660685962059142644, 2.91893768888198608697520250989, 3.82813941873210653378196391269, 4.75959099094212607849065127461, 5.86637308739081627632196487749, 6.09941866119697653779583060823, 7.27506634509109281592273390108, 7.67106396526759761595204212560, 8.429873255960323474871716302272