| L(s) = 1 | − 4.24·5-s + 1.41·13-s − 8·17-s + 12.9·25-s + 9.89·29-s + 7.07·37-s − 8·41-s − 7·49-s + 7.07·53-s − 1.41·61-s − 6·65-s + 6·73-s + 33.9·85-s + 10·89-s + 8·97-s + 15.5·101-s + 9.89·109-s − 14·113-s + ⋯ |
| L(s) = 1 | − 1.89·5-s + 0.392·13-s − 1.94·17-s + 2.59·25-s + 1.83·29-s + 1.16·37-s − 1.24·41-s − 49-s + 0.971·53-s − 0.181·61-s − 0.744·65-s + 0.702·73-s + 3.68·85-s + 1.05·89-s + 0.812·97-s + 1.54·101-s + 0.948·109-s − 1.31·113-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 | \( 1 + 4.24T + 5T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 1.41T + 13T^{2} \) |
| 17 | \( 1 + 8T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 9.89T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 7.07T + 37T^{2} \) |
| 41 | \( 1 + 8T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 7.07T + 53T^{2} \) |
| 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 | \( 1 - 10T + 89T^{2} \) |
| 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
−7.44043345892228247149549323760, −6.68684263495941383046572458692, −6.33593150064612314381246820711, −4.95569091972956268503452373605, −4.55032572487689525854142562697, −3.90065321411754520542984845483, −3.18735062372663579821334436814, −2.33306822173763324034577501888, −0.959013378190695280656212346282, 0,
0.959013378190695280656212346282, 2.33306822173763324034577501888, 3.18735062372663579821334436814, 3.90065321411754520542984845483, 4.55032572487689525854142562697, 4.95569091972956268503452373605, 6.33593150064612314381246820711, 6.68684263495941383046572458692, 7.44043345892228247149549323760
