| L(s) = 1 | − 4·2-s + 3·3-s + 8·4-s − 12·6-s + 21·7-s + 9·9-s + 11·11-s + 24·12-s − 68·13-s − 84·14-s − 64·16-s + 21·17-s − 36·18-s + 125·19-s + 63·21-s − 44·22-s + 137·23-s + 272·26-s + 27·27-s + 168·28-s − 150·29-s + 292·31-s + 256·32-s + 33·33-s − 84·34-s + 72·36-s − 349·37-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 0.577·3-s + 4-s − 0.816·6-s + 1.13·7-s + 1/3·9-s + 0.301·11-s + 0.577·12-s − 1.45·13-s − 1.60·14-s − 16-s + 0.299·17-s − 0.471·18-s + 1.50·19-s + 0.654·21-s − 0.426·22-s + 1.24·23-s + 2.05·26-s + 0.192·27-s + 1.13·28-s − 0.960·29-s + 1.69·31-s + 1.41·32-s + 0.174·33-s − 0.423·34-s + 1/3·36-s − 1.55·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.442037240\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.442037240\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - p T \) |
| 5 | \( 1 \) |
| 11 | \( 1 - p T \) |
| good | 2 | \( 1 + p^{2} T + p^{3} T^{2} \) |
| 7 | \( 1 - 3 p T + p^{3} T^{2} \) |
| 13 | \( 1 + 68 T + p^{3} T^{2} \) |
| 17 | \( 1 - 21 T + p^{3} T^{2} \) |
| 19 | \( 1 - 125 T + p^{3} T^{2} \) |
| 23 | \( 1 - 137 T + p^{3} T^{2} \) |
| 29 | \( 1 + 150 T + p^{3} T^{2} \) |
| 31 | \( 1 - 292 T + p^{3} T^{2} \) |
| 37 | \( 1 + 349 T + p^{3} T^{2} \) |
| 41 | \( 1 - 497 T + p^{3} T^{2} \) |
| 43 | \( 1 + 208 T + p^{3} T^{2} \) |
| 47 | \( 1 + 369 T + p^{3} T^{2} \) |
| 53 | \( 1 - 542 T + p^{3} T^{2} \) |
| 59 | \( 1 - 235 T + p^{3} T^{2} \) |
| 61 | \( 1 - 482 T + p^{3} T^{2} \) |
| 67 | \( 1 + 734 T + p^{3} T^{2} \) |
| 71 | \( 1 - 587 T + p^{3} T^{2} \) |
| 73 | \( 1 + 518 T + p^{3} T^{2} \) |
| 79 | \( 1 + 1045 T + p^{3} T^{2} \) |
| 83 | \( 1 + 608 T + p^{3} T^{2} \) |
| 89 | \( 1 + 770 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1541 T + p^{3} T^{2} \) |
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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
−9.735570357369297719355795089800, −8.972134561368547237823715843351, −8.246699748276965308771467325843, −7.45383172789735110046698459376, −7.05810308114537768581064304355, −5.30773150647082039720165442511, −4.50454835370355212269610742031, −2.94223057474751527672352047618, −1.81562244915030458618086377596, −0.848299949214220174726968276092,
0.848299949214220174726968276092, 1.81562244915030458618086377596, 2.94223057474751527672352047618, 4.50454835370355212269610742031, 5.30773150647082039720165442511, 7.05810308114537768581064304355, 7.45383172789735110046698459376, 8.246699748276965308771467325843, 8.972134561368547237823715843351, 9.735570357369297719355795089800
