| L(s) = 1 | + 2.41·2-s + 3.82·4-s − 3.41·7-s + 4.41·8-s + 1.41·11-s − 2.58·13-s − 8.24·14-s + 2.99·16-s − 6.82·17-s + 19-s + 3.41·22-s − 3.65·23-s − 6.24·26-s − 13.0·28-s − 5.07·29-s − 10.4·31-s − 1.58·32-s − 16.4·34-s + 3.07·37-s + 2.41·38-s + 4.58·41-s − 3.41·43-s + 5.41·44-s − 8.82·46-s + 11.6·47-s + 4.65·49-s − 9.89·52-s + ⋯ |
| L(s) = 1 | + 1.70·2-s + 1.91·4-s − 1.29·7-s + 1.56·8-s + 0.426·11-s − 0.717·13-s − 2.20·14-s + 0.749·16-s − 1.65·17-s + 0.229·19-s + 0.727·22-s − 0.762·23-s − 1.22·26-s − 2.47·28-s − 0.941·29-s − 1.88·31-s − 0.280·32-s − 2.82·34-s + 0.504·37-s + 0.391·38-s + 0.716·41-s − 0.520·43-s + 0.816·44-s − 1.30·46-s + 1.70·47-s + 0.665·49-s − 1.37·52-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 | \( 1 - 2.41T + 2T^{2} \) |
| 7 | \( 1 + 3.41T + 7T^{2} \) |
| 11 | \( 1 - 1.41T + 11T^{2} \) |
| 13 | \( 1 + 2.58T + 13T^{2} \) |
| 17 | \( 1 + 6.82T + 17T^{2} \) |
| 23 | \( 1 + 3.65T + 23T^{2} \) |
| 29 | \( 1 + 5.07T + 29T^{2} \) |
| 31 | \( 1 + 10.4T + 31T^{2} \) |
| 37 | \( 1 - 3.07T + 37T^{2} \) |
| 41 | \( 1 - 4.58T + 41T^{2} \) |
| 43 | \( 1 + 3.41T + 43T^{2} \) |
| 47 | \( 1 - 11.6T + 47T^{2} \) |
| 53 | \( 1 - 4T + 53T^{2} \) |
| 59 | \( 1 - 8.48T + 59T^{2} \) |
| 61 | \( 1 + 5.65T + 61T^{2} \) |
| 67 | \( 1 + 12T + 67T^{2} \) |
| 71 | \( 1 + 12.4T + 71T^{2} \) |
| 73 | \( 1 - 2T + 73T^{2} \) |
| 79 | \( 1 - 11.3T + 79T^{2} \) |
| 83 | \( 1 - 6.48T + 83T^{2} \) |
| 89 | \( 1 - 14.7T + 89T^{2} \) |
| 97 | \( 1 + 4.24T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.51115303589102797161468777280, −7.06815547519602159872205392482, −6.28748234578207812488485428215, −5.84170730043084026177542663517, −4.97596443820805829007042907843, −4.08175854079333487749314144381, −3.67315497872995440881707885943, −2.67751703285982928686704922668, −2.01799075059631958924524305554, 0,
2.01799075059631958924524305554, 2.67751703285982928686704922668, 3.67315497872995440881707885943, 4.08175854079333487749314144381, 4.97596443820805829007042907843, 5.84170730043084026177542663517, 6.28748234578207812488485428215, 7.06815547519602159872205392482, 7.51115303589102797161468777280
