| L(s) = 1 | − 2-s + 4-s − 7-s − 8-s − 3·9-s − 11-s − 13-s + 14-s + 16-s + 6·17-s + 3·18-s + 19-s + 22-s + 23-s + 26-s − 28-s − 5·29-s − 7·31-s − 32-s − 6·34-s − 3·36-s + 8·37-s − 38-s + 3·43-s − 44-s − 46-s + 49-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.377·7-s − 0.353·8-s − 9-s − 0.301·11-s − 0.277·13-s + 0.267·14-s + 1/4·16-s + 1.45·17-s + 0.707·18-s + 0.229·19-s + 0.213·22-s + 0.208·23-s + 0.196·26-s − 0.188·28-s − 0.928·29-s − 1.25·31-s − 0.176·32-s − 1.02·34-s − 1/2·36-s + 1.31·37-s − 0.162·38-s + 0.457·43-s − 0.150·44-s − 0.147·46-s + 1/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9309346682\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9309346682\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 3 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - T + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 - 13 T + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 97 | \( 1 - 13 T + p T^{2} \) |
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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
−8.519993406514513200992758977147, −7.65503686209496864274965980737, −7.38874352240161126760348797775, −6.14416075045374234152869573562, −5.75195799175156518564650697212, −4.86460322633326441536297292478, −3.52521011547767926993332959115, −2.96905260568813613797727258028, −1.91638172028453614829966821720, −0.60299685439283961232608096588,
0.60299685439283961232608096588, 1.91638172028453614829966821720, 2.96905260568813613797727258028, 3.52521011547767926993332959115, 4.86460322633326441536297292478, 5.75195799175156518564650697212, 6.14416075045374234152869573562, 7.38874352240161126760348797775, 7.65503686209496864274965980737, 8.519993406514513200992758977147
