L(s) = 1 | − 2-s + 4-s + 2·5-s − 8-s − 2·10-s + 11-s − 5.41·13-s + 16-s + 2·17-s + 2.24·19-s + 2·20-s − 22-s + 4.82·23-s − 25-s + 5.41·26-s − 9.65·29-s − 6.24·31-s − 32-s − 2·34-s + 0.828·37-s − 2.24·38-s − 2·40-s + 11.6·41-s − 4.82·43-s + 44-s − 4.82·46-s − 7.89·47-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.5·4-s + 0.894·5-s − 0.353·8-s − 0.632·10-s + 0.301·11-s − 1.50·13-s + 0.250·16-s + 0.485·17-s + 0.514·19-s + 0.447·20-s − 0.213·22-s + 1.00·23-s − 0.200·25-s + 1.06·26-s − 1.79·29-s − 1.12·31-s − 0.176·32-s − 0.342·34-s + 0.136·37-s − 0.363·38-s − 0.316·40-s + 1.82·41-s − 0.736·43-s + 0.150·44-s − 0.711·46-s − 1.15·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
good | 5 | \( 1 - 2T + 5T^{2} \) |
| 13 | \( 1 + 5.41T + 13T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 19 | \( 1 - 2.24T + 19T^{2} \) |
| 23 | \( 1 - 4.82T + 23T^{2} \) |
| 29 | \( 1 + 9.65T + 29T^{2} \) |
| 31 | \( 1 + 6.24T + 31T^{2} \) |
| 37 | \( 1 - 0.828T + 37T^{2} \) |
| 41 | \( 1 - 11.6T + 41T^{2} \) |
| 43 | \( 1 + 4.82T + 43T^{2} \) |
| 47 | \( 1 + 7.89T + 47T^{2} \) |
| 53 | \( 1 - 10.4T + 53T^{2} \) |
| 59 | \( 1 - 6.82T + 59T^{2} \) |
| 61 | \( 1 + 2.58T + 61T^{2} \) |
| 67 | \( 1 + 1.17T + 67T^{2} \) |
| 71 | \( 1 + 5.65T + 71T^{2} \) |
| 73 | \( 1 + 11.6T + 73T^{2} \) |
| 79 | \( 1 + 16.4T + 79T^{2} \) |
| 83 | \( 1 - 5.07T + 83T^{2} \) |
| 89 | \( 1 - 3.75T + 89T^{2} \) |
| 97 | \( 1 - 9.89T + 97T^{2} \) |
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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
−7.25400396987864757599306270862, −6.99381800976401536451564333563, −5.79072982294630856807743882864, −5.59135440786651729989460371137, −4.69426683598062010999446331213, −3.66631857556829425311793961539, −2.77129485627538823884269245893, −2.06395870865954971515008918928, −1.27413014813717384512932744551, 0,
1.27413014813717384512932744551, 2.06395870865954971515008918928, 2.77129485627538823884269245893, 3.66631857556829425311793961539, 4.69426683598062010999446331213, 5.59135440786651729989460371137, 5.79072982294630856807743882864, 6.99381800976401536451564333563, 7.25400396987864757599306270862