| L(s) = 1 | − 2-s + 4-s − 7-s − 8-s − 3·9-s − 4·11-s − 4·13-s + 14-s + 16-s − 4·17-s + 3·18-s − 4·19-s + 4·22-s + 23-s − 5·25-s + 4·26-s − 28-s − 29-s − 2·31-s − 32-s + 4·34-s − 3·36-s − 10·37-s + 4·38-s + 4·41-s + 4·43-s − 4·44-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.377·7-s − 0.353·8-s − 9-s − 1.20·11-s − 1.10·13-s + 0.267·14-s + 1/4·16-s − 0.970·17-s + 0.707·18-s − 0.917·19-s + 0.852·22-s + 0.208·23-s − 25-s + 0.784·26-s − 0.188·28-s − 0.185·29-s − 0.359·31-s − 0.176·32-s + 0.685·34-s − 1/2·36-s − 1.64·37-s + 0.648·38-s + 0.624·41-s + 0.609·43-s − 0.603·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9338 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9338 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 10 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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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
−7.12624546031253533937831412650, −6.45860314269040325315289351402, −5.69991488583671424244505958864, −5.10988157116511824690321350846, −4.22040684244739727400804742168, −3.13314428479138335146586499047, −2.52235473228463594328055110468, −1.86453548997989284712573400416, 0, 0,
1.86453548997989284712573400416, 2.52235473228463594328055110468, 3.13314428479138335146586499047, 4.22040684244739727400804742168, 5.10988157116511824690321350846, 5.69991488583671424244505958864, 6.45860314269040325315289351402, 7.12624546031253533937831412650
