| L(s) = 1 | + 0.613·3-s − 2.62·9-s + 11-s − 5.29·13-s − 5.90·17-s + 7.13·19-s − 2.62·23-s − 3.45·27-s + 4.62·29-s + 6.51·31-s + 0.613·33-s + 2.62·37-s − 3.24·39-s − 1.84·41-s − 1.37·43-s + 1.22·47-s − 3.62·51-s − 9.24·53-s + 4.37·57-s − 5.29·59-s − 7.74·61-s + 12.2·67-s − 1.60·69-s + 12.6·71-s − 3.06·73-s − 4.62·79-s + 5.75·81-s + ⋯ |
| L(s) = 1 | + 0.354·3-s − 0.874·9-s + 0.301·11-s − 1.46·13-s − 1.43·17-s + 1.63·19-s − 0.547·23-s − 0.664·27-s + 0.858·29-s + 1.17·31-s + 0.106·33-s + 0.431·37-s − 0.519·39-s − 0.287·41-s − 0.209·43-s + 0.179·47-s − 0.507·51-s − 1.27·53-s + 0.579·57-s − 0.688·59-s − 0.991·61-s + 1.49·67-s − 0.193·69-s + 1.49·71-s − 0.359·73-s − 0.520·79-s + 0.639·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.600478802\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.600478802\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - 0.613T + 3T^{2} \) |
| 11 | \( 1 - T + 11T^{2} \) |
| 13 | \( 1 + 5.29T + 13T^{2} \) |
| 17 | \( 1 + 5.90T + 17T^{2} \) |
| 19 | \( 1 - 7.13T + 19T^{2} \) |
| 23 | \( 1 + 2.62T + 23T^{2} \) |
| 29 | \( 1 - 4.62T + 29T^{2} \) |
| 31 | \( 1 - 6.51T + 31T^{2} \) |
| 37 | \( 1 - 2.62T + 37T^{2} \) |
| 41 | \( 1 + 1.84T + 41T^{2} \) |
| 43 | \( 1 + 1.37T + 43T^{2} \) |
| 47 | \( 1 - 1.22T + 47T^{2} \) |
| 53 | \( 1 + 9.24T + 53T^{2} \) |
| 59 | \( 1 + 5.29T + 59T^{2} \) |
| 61 | \( 1 + 7.74T + 61T^{2} \) |
| 67 | \( 1 - 12.2T + 67T^{2} \) |
| 71 | \( 1 - 12.6T + 71T^{2} \) |
| 73 | \( 1 + 3.06T + 73T^{2} \) |
| 79 | \( 1 + 4.62T + 79T^{2} \) |
| 83 | \( 1 - 12.4T + 83T^{2} \) |
| 89 | \( 1 + 12.4T + 89T^{2} \) |
| 97 | \( 1 - 8.97T + 97T^{2} \) |
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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
−7.88089145355976785401138823553, −6.92577319561101327127236086141, −6.45135636590207723228668486424, −5.55885553723488377938029428931, −4.87829158053605709234198019784, −4.28813604698553195848793229041, −3.19187988723790917264305152506, −2.69766351015186077194525684041, −1.89233884989080443203199131605, −0.56544740279164568365905722591,
0.56544740279164568365905722591, 1.89233884989080443203199131605, 2.69766351015186077194525684041, 3.19187988723790917264305152506, 4.28813604698553195848793229041, 4.87829158053605709234198019784, 5.55885553723488377938029428931, 6.45135636590207723228668486424, 6.92577319561101327127236086141, 7.88089145355976785401138823553
