L(s) = 1 | − 0.139·2-s − 2.98·3-s − 1.98·4-s + 0.416·6-s − 3.56·7-s + 0.554·8-s + 5.91·9-s + 5.91·12-s − 1.19·13-s + 0.497·14-s + 3.88·16-s − 5.43·17-s − 0.824·18-s − 4.80·19-s + 10.6·21-s + 1.82·23-s − 1.65·24-s + 0.166·26-s − 8.70·27-s + 7.06·28-s + 4.14·29-s + 1.28·31-s − 1.65·32-s + 0.757·34-s − 11.7·36-s + 3.16·37-s + 0.669·38-s + ⋯ |
L(s) = 1 | − 0.0985·2-s − 1.72·3-s − 0.990·4-s + 0.169·6-s − 1.34·7-s + 0.196·8-s + 1.97·9-s + 1.70·12-s − 0.331·13-s + 0.132·14-s + 0.970·16-s − 1.31·17-s − 0.194·18-s − 1.10·19-s + 2.32·21-s + 0.379·23-s − 0.338·24-s + 0.0326·26-s − 1.67·27-s + 1.33·28-s + 0.769·29-s + 0.231·31-s − 0.291·32-s + 0.129·34-s − 1.95·36-s + 0.519·37-s + 0.108·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + 0.139T + 2T^{2} \) |
| 3 | \( 1 + 2.98T + 3T^{2} \) |
| 7 | \( 1 + 3.56T + 7T^{2} \) |
| 13 | \( 1 + 1.19T + 13T^{2} \) |
| 17 | \( 1 + 5.43T + 17T^{2} \) |
| 19 | \( 1 + 4.80T + 19T^{2} \) |
| 23 | \( 1 - 1.82T + 23T^{2} \) |
| 29 | \( 1 - 4.14T + 29T^{2} \) |
| 31 | \( 1 - 1.28T + 31T^{2} \) |
| 37 | \( 1 - 3.16T + 37T^{2} \) |
| 41 | \( 1 - 6.40T + 41T^{2} \) |
| 43 | \( 1 - 4.23T + 43T^{2} \) |
| 47 | \( 1 - 7.02T + 47T^{2} \) |
| 53 | \( 1 - 10.8T + 53T^{2} \) |
| 59 | \( 1 + 4.01T + 59T^{2} \) |
| 61 | \( 1 - 14.4T + 61T^{2} \) |
| 67 | \( 1 + 8.14T + 67T^{2} \) |
| 71 | \( 1 - 8.51T + 71T^{2} \) |
| 73 | \( 1 + 8.54T + 73T^{2} \) |
| 79 | \( 1 + 0.393T + 79T^{2} \) |
| 83 | \( 1 + 1.56T + 83T^{2} \) |
| 89 | \( 1 + 6.90T + 89T^{2} \) |
| 97 | \( 1 - 9.31T + 97T^{2} \) |
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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
−8.537573570405541468816929784039, −7.31358885848646721098260067117, −6.62250034016318554863835996661, −6.07404534408812514240356011634, −5.35159165335360912906972208689, −4.43468662491240233372821820450, −4.02640419884882957054479232193, −2.56602590309550477279535029345, −0.838427768899319379822473147152, 0,
0.838427768899319379822473147152, 2.56602590309550477279535029345, 4.02640419884882957054479232193, 4.43468662491240233372821820450, 5.35159165335360912906972208689, 6.07404534408812514240356011634, 6.62250034016318554863835996661, 7.31358885848646721098260067117, 8.537573570405541468816929784039