| L(s) = 1 | − 0.754·2-s − 3-s − 1.43·4-s + 0.754·6-s − 4.18·7-s + 2.58·8-s + 9-s + 0.596·11-s + 1.43·12-s + 2.18·13-s + 3.15·14-s + 0.908·16-s − 2.81·17-s − 0.754·18-s + 0.528·19-s + 4.18·21-s − 0.450·22-s − 0.590·23-s − 2.58·24-s − 1.64·26-s − 27-s + 5.98·28-s + 29-s + 8.02·31-s − 5.86·32-s − 0.596·33-s + 2.12·34-s + ⋯ |
| L(s) = 1 | − 0.533·2-s − 0.577·3-s − 0.715·4-s + 0.308·6-s − 1.58·7-s + 0.915·8-s + 0.333·9-s + 0.179·11-s + 0.413·12-s + 0.606·13-s + 0.843·14-s + 0.227·16-s − 0.682·17-s − 0.177·18-s + 0.121·19-s + 0.913·21-s − 0.0960·22-s − 0.123·23-s − 0.528·24-s − 0.323·26-s − 0.192·27-s + 1.13·28-s + 0.185·29-s + 1.44·31-s − 1.03·32-s − 0.103·33-s + 0.363·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 + 0.754T + 2T^{2} \) |
| 7 | \( 1 + 4.18T + 7T^{2} \) |
| 11 | \( 1 - 0.596T + 11T^{2} \) |
| 13 | \( 1 - 2.18T + 13T^{2} \) |
| 17 | \( 1 + 2.81T + 17T^{2} \) |
| 19 | \( 1 - 0.528T + 19T^{2} \) |
| 23 | \( 1 + 0.590T + 23T^{2} \) |
| 31 | \( 1 - 8.02T + 31T^{2} \) |
| 37 | \( 1 - 2.52T + 37T^{2} \) |
| 41 | \( 1 - 1.57T + 41T^{2} \) |
| 43 | \( 1 - 6.98T + 43T^{2} \) |
| 47 | \( 1 - 2.57T + 47T^{2} \) |
| 53 | \( 1 + 10.3T + 53T^{2} \) |
| 59 | \( 1 - 13.9T + 59T^{2} \) |
| 61 | \( 1 + 11.1T + 61T^{2} \) |
| 67 | \( 1 + 0.614T + 67T^{2} \) |
| 71 | \( 1 - 9.28T + 71T^{2} \) |
| 73 | \( 1 + 9.59T + 73T^{2} \) |
| 79 | \( 1 + 10.3T + 79T^{2} \) |
| 83 | \( 1 + 3.92T + 83T^{2} \) |
| 89 | \( 1 + 12.3T + 89T^{2} \) |
| 97 | \( 1 + 3.21T + 97T^{2} \) |
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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
−8.859526112336300134003478691936, −8.032906487908560067729109043697, −7.04030914180551224372856395710, −6.35280605799936789759369877241, −5.67977854501064474690570106947, −4.53246181097811265902315628208, −3.86896209949477103890157631348, −2.77032007654021612627617461231, −1.11834923788114409681819649097, 0,
1.11834923788114409681819649097, 2.77032007654021612627617461231, 3.86896209949477103890157631348, 4.53246181097811265902315628208, 5.67977854501064474690570106947, 6.35280605799936789759369877241, 7.04030914180551224372856395710, 8.032906487908560067729109043697, 8.859526112336300134003478691936
