L(s) = 1 | − 1.41·5-s − 1.41·13-s − 2·17-s − 2.99·25-s + 4.24·29-s + 7.07·37-s + 10·41-s − 7·49-s + 12.7·53-s − 1.41·61-s + 2.00·65-s − 6·73-s + 2.82·85-s − 16·89-s − 8·97-s − 12.7·101-s − 9.89·109-s − 16·113-s + ⋯ |
L(s) = 1 | − 0.632·5-s − 0.392·13-s − 0.485·17-s − 0.599·25-s + 0.787·29-s + 1.16·37-s + 1.56·41-s − 49-s + 1.74·53-s − 0.181·61-s + 0.248·65-s − 0.702·73-s + 0.306·85-s − 1.69·89-s − 0.812·97-s − 1.26·101-s − 0.948·109-s − 1.50·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + 1.41T + 5T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 1.41T + 13T^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 4.24T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 7.07T + 37T^{2} \) |
| 41 | \( 1 - 10T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 12.7T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 1.41T + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 6T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + 16T + 89T^{2} \) |
| 97 | \( 1 + 8T + 97T^{2} \) |
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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
−7.41367203544017179150150374496, −6.78180926260612244285230409721, −6.02178589114613659827561760407, −5.32156971542222217447771814393, −4.36951863053797425206194896841, −4.05812315348108682270022298463, −2.97354746376572567632804321562, −2.32258618132997454349170008553, −1.12962382380261218989277859979, 0,
1.12962382380261218989277859979, 2.32258618132997454349170008553, 2.97354746376572567632804321562, 4.05812315348108682270022298463, 4.36951863053797425206194896841, 5.32156971542222217447771814393, 6.02178589114613659827561760407, 6.78180926260612244285230409721, 7.41367203544017179150150374496