| L(s) = 1 | + 2.36·2-s + 3.57·4-s − 0.784·7-s + 3.72·8-s − 2.93·11-s − 4·13-s − 1.85·14-s + 1.63·16-s − 5.15·17-s − 19-s − 6.93·22-s − 3.78·23-s − 9.44·26-s − 2.80·28-s + 5·29-s + 1.06·31-s − 3.57·32-s − 12.1·34-s + 0.722·37-s − 2.36·38-s − 7.93·41-s − 4·43-s − 10.5·44-s − 8.93·46-s − 3.87·47-s − 6.38·49-s − 14.3·52-s + ⋯ |
| L(s) = 1 | + 1.66·2-s + 1.78·4-s − 0.296·7-s + 1.31·8-s − 0.885·11-s − 1.10·13-s − 0.495·14-s + 0.409·16-s − 1.24·17-s − 0.229·19-s − 1.47·22-s − 0.789·23-s − 1.85·26-s − 0.530·28-s + 0.928·29-s + 0.190·31-s − 0.632·32-s − 2.08·34-s + 0.118·37-s − 0.383·38-s − 1.23·41-s − 0.609·43-s − 1.58·44-s − 1.31·46-s − 0.565·47-s − 0.911·49-s − 1.98·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4275 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 - 2.36T + 2T^{2} \) |
| 7 | \( 1 + 0.784T + 7T^{2} \) |
| 11 | \( 1 + 2.93T + 11T^{2} \) |
| 13 | \( 1 + 4T + 13T^{2} \) |
| 17 | \( 1 + 5.15T + 17T^{2} \) |
| 23 | \( 1 + 3.78T + 23T^{2} \) |
| 29 | \( 1 - 5T + 29T^{2} \) |
| 31 | \( 1 - 1.06T + 31T^{2} \) |
| 37 | \( 1 - 0.722T + 37T^{2} \) |
| 41 | \( 1 + 7.93T + 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 + 3.87T + 47T^{2} \) |
| 53 | \( 1 - 6.44T + 53T^{2} \) |
| 59 | \( 1 - 4.66T + 59T^{2} \) |
| 61 | \( 1 - 8.87T + 61T^{2} \) |
| 67 | \( 1 + 8.93T + 67T^{2} \) |
| 71 | \( 1 - 2.66T + 71T^{2} \) |
| 73 | \( 1 - 8.44T + 73T^{2} \) |
| 79 | \( 1 - 3.35T + 79T^{2} \) |
| 83 | \( 1 - 4.93T + 83T^{2} \) |
| 89 | \( 1 + 0.569T + 89T^{2} \) |
| 97 | \( 1 + 8.59T + 97T^{2} \) |
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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
−7.84373162863912574932877542224, −6.82845868755271393030182908296, −6.56941720787635864575756749427, −5.56872325025590433515194561882, −4.93946596039844254161023269146, −4.40292486845125305721525921109, −3.48586444181951134127886006135, −2.62547216699874871537690867155, −2.06063082697105704858324256726, 0,
2.06063082697105704858324256726, 2.62547216699874871537690867155, 3.48586444181951134127886006135, 4.40292486845125305721525921109, 4.93946596039844254161023269146, 5.56872325025590433515194561882, 6.56941720787635864575756749427, 6.82845868755271393030182908296, 7.84373162863912574932877542224
