| L(s) = 1 | − 2-s + 4-s − 1.23·7-s − 8-s − 2·11-s + 5.23·13-s + 1.23·14-s + 16-s − 17-s − 6.47·19-s + 2·22-s − 4·23-s − 5.23·26-s − 1.23·28-s + 4·29-s + 2.47·31-s − 32-s + 34-s + 2·37-s + 6.47·38-s + 7.70·41-s + 12.1·43-s − 2·44-s + 4·46-s − 10.4·47-s − 5.47·49-s + 5.23·52-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 0.467·7-s − 0.353·8-s − 0.603·11-s + 1.45·13-s + 0.330·14-s + 0.250·16-s − 0.242·17-s − 1.48·19-s + 0.426·22-s − 0.834·23-s − 1.02·26-s − 0.233·28-s + 0.742·29-s + 0.444·31-s − 0.176·32-s + 0.171·34-s + 0.328·37-s + 1.04·38-s + 1.20·41-s + 1.85·43-s − 0.301·44-s + 0.589·46-s − 1.52·47-s − 0.781·49-s + 0.726·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 7 | \( 1 + 1.23T + 7T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 - 5.23T + 13T^{2} \) |
| 19 | \( 1 + 6.47T + 19T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 - 4T + 29T^{2} \) |
| 31 | \( 1 - 2.47T + 31T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 - 7.70T + 41T^{2} \) |
| 43 | \( 1 - 12.1T + 43T^{2} \) |
| 47 | \( 1 + 10.4T + 47T^{2} \) |
| 53 | \( 1 - 10.9T + 53T^{2} \) |
| 59 | \( 1 + 11.7T + 59T^{2} \) |
| 61 | \( 1 - 4.47T + 61T^{2} \) |
| 67 | \( 1 + 1.70T + 67T^{2} \) |
| 71 | \( 1 + 11.2T + 71T^{2} \) |
| 73 | \( 1 - 8.76T + 73T^{2} \) |
| 79 | \( 1 - 0.944T + 79T^{2} \) |
| 83 | \( 1 + 10.4T + 83T^{2} \) |
| 89 | \( 1 + 12.4T + 89T^{2} \) |
| 97 | \( 1 + 1.70T + 97T^{2} \) |
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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
−7.70682967747285430763690833760, −6.79626633687161792939813202185, −6.16680290288510962891099299414, −5.80970956771468242324540177221, −4.54595824187663315757080607331, −3.91944101663760734989432064310, −2.93620198046504603677146054564, −2.20489717210985408408596030118, −1.14248795719803245422624664248, 0,
1.14248795719803245422624664248, 2.20489717210985408408596030118, 2.93620198046504603677146054564, 3.91944101663760734989432064310, 4.54595824187663315757080607331, 5.80970956771468242324540177221, 6.16680290288510962891099299414, 6.79626633687161792939813202185, 7.70682967747285430763690833760
