L(s) = 1 | − 2.54·2-s − 3-s + 4.46·4-s + 4.07·5-s + 2.54·6-s + 7-s − 6.26·8-s + 9-s − 10.3·10-s − 3.87·11-s − 4.46·12-s − 2.54·14-s − 4.07·15-s + 6.99·16-s − 2.47·17-s − 2.54·18-s − 2.67·19-s + 18.1·20-s − 21-s + 9.85·22-s − 4.84·23-s + 6.26·24-s + 11.5·25-s − 27-s + 4.46·28-s + 5.76·29-s + 10.3·30-s + ⋯ |
L(s) = 1 | − 1.79·2-s − 0.577·3-s + 2.23·4-s + 1.82·5-s + 1.03·6-s + 0.377·7-s − 2.21·8-s + 0.333·9-s − 3.27·10-s − 1.16·11-s − 1.28·12-s − 0.679·14-s − 1.05·15-s + 1.74·16-s − 0.599·17-s − 0.599·18-s − 0.614·19-s + 4.06·20-s − 0.218·21-s + 2.10·22-s − 1.01·23-s + 1.27·24-s + 2.31·25-s − 0.192·27-s + 0.843·28-s + 1.07·29-s + 1.89·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + 2.54T + 2T^{2} \) |
| 5 | \( 1 - 4.07T + 5T^{2} \) |
| 11 | \( 1 + 3.87T + 11T^{2} \) |
| 17 | \( 1 + 2.47T + 17T^{2} \) |
| 19 | \( 1 + 2.67T + 19T^{2} \) |
| 23 | \( 1 + 4.84T + 23T^{2} \) |
| 29 | \( 1 - 5.76T + 29T^{2} \) |
| 31 | \( 1 + 4.35T + 31T^{2} \) |
| 37 | \( 1 - 2.25T + 37T^{2} \) |
| 41 | \( 1 + 4.84T + 41T^{2} \) |
| 43 | \( 1 - 3.69T + 43T^{2} \) |
| 47 | \( 1 + 10.0T + 47T^{2} \) |
| 53 | \( 1 - 11.2T + 53T^{2} \) |
| 59 | \( 1 + 8.39T + 59T^{2} \) |
| 61 | \( 1 + 7.02T + 61T^{2} \) |
| 67 | \( 1 - 5.13T + 67T^{2} \) |
| 71 | \( 1 + 9.80T + 71T^{2} \) |
| 73 | \( 1 + 8.54T + 73T^{2} \) |
| 79 | \( 1 - 0.0251T + 79T^{2} \) |
| 83 | \( 1 + 0.202T + 83T^{2} \) |
| 89 | \( 1 + 18.2T + 89T^{2} \) |
| 97 | \( 1 - 10.6T + 97T^{2} \) |
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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
−8.406299798252270440052286221379, −7.59256299995369470414112212991, −6.75521177692379661903124525600, −6.15898545482365672135125173550, −5.53860544384402963934619855924, −4.63156941643665710935143557959, −2.75616381590470253747192036956, −2.09049313149949409889891557662, −1.38364644972178289219956954558, 0,
1.38364644972178289219956954558, 2.09049313149949409889891557662, 2.75616381590470253747192036956, 4.63156941643665710935143557959, 5.53860544384402963934619855924, 6.15898545482365672135125173550, 6.75521177692379661903124525600, 7.59256299995369470414112212991, 8.406299798252270440052286221379