L(s) = 1 | − 0.634·2-s − 1.59·4-s − 2.74·7-s + 2.28·8-s − 4.00·11-s + 13-s + 1.74·14-s + 1.74·16-s + 2.35·17-s + 4.69·19-s + 2.54·22-s + 1.88·23-s − 0.634·26-s + 4.39·28-s + 3.19·31-s − 5.67·32-s − 1.49·34-s − 6.44·37-s − 2.97·38-s + 11.5·41-s + 0.691·43-s + 6.40·44-s − 1.19·46-s − 5.89·47-s + 0.552·49-s − 1.59·52-s + 9.14·53-s + ⋯ |
L(s) = 1 | − 0.448·2-s − 0.798·4-s − 1.03·7-s + 0.806·8-s − 1.20·11-s + 0.277·13-s + 0.465·14-s + 0.437·16-s + 0.572·17-s + 1.07·19-s + 0.541·22-s + 0.393·23-s − 0.124·26-s + 0.829·28-s + 0.573·31-s − 1.00·32-s − 0.256·34-s − 1.05·37-s − 0.482·38-s + 1.80·41-s + 0.105·43-s + 0.965·44-s − 0.176·46-s − 0.859·47-s + 0.0789·49-s − 0.221·52-s + 1.25·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 - T \) |
good | 2 | \( 1 + 0.634T + 2T^{2} \) |
| 7 | \( 1 + 2.74T + 7T^{2} \) |
| 11 | \( 1 + 4.00T + 11T^{2} \) |
| 17 | \( 1 - 2.35T + 17T^{2} \) |
| 19 | \( 1 - 4.69T + 19T^{2} \) |
| 23 | \( 1 - 1.88T + 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 3.19T + 31T^{2} \) |
| 37 | \( 1 + 6.44T + 37T^{2} \) |
| 41 | \( 1 - 11.5T + 41T^{2} \) |
| 43 | \( 1 - 0.691T + 43T^{2} \) |
| 47 | \( 1 + 5.89T + 47T^{2} \) |
| 53 | \( 1 - 9.14T + 53T^{2} \) |
| 59 | \( 1 - 1.17T + 59T^{2} \) |
| 61 | \( 1 + 4.44T + 61T^{2} \) |
| 67 | \( 1 + 4.80T + 67T^{2} \) |
| 71 | \( 1 - 1.82T + 71T^{2} \) |
| 73 | \( 1 + 15.3T + 73T^{2} \) |
| 79 | \( 1 + 10.6T + 79T^{2} \) |
| 83 | \( 1 + 8.42T + 83T^{2} \) |
| 89 | \( 1 + 5.75T + 89T^{2} \) |
| 97 | \( 1 + 11.9T + 97T^{2} \) |
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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
−8.484401251821378861154016836758, −7.66034127625577950017181173873, −7.13811276393380168651029618346, −5.94844875184154030825830465661, −5.36269270719890828333500426071, −4.48241282904276873373670103430, −3.46008775977508702403999237501, −2.75670996760833194461812164664, −1.17481167966041735397887190724, 0,
1.17481167966041735397887190724, 2.75670996760833194461812164664, 3.46008775977508702403999237501, 4.48241282904276873373670103430, 5.36269270719890828333500426071, 5.94844875184154030825830465661, 7.13811276393380168651029618346, 7.66034127625577950017181173873, 8.484401251821378861154016836758