L(s) = 1 | + 2-s + 4-s − 7-s + 8-s − 2·11-s − 4·13-s − 14-s + 16-s − 19-s − 2·22-s − 2·23-s − 5·25-s − 4·26-s − 28-s − 6·29-s − 4·31-s + 32-s − 6·37-s − 38-s + 6·41-s − 2·44-s − 2·46-s + 4·47-s + 49-s − 5·50-s − 4·52-s + 10·53-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.377·7-s + 0.353·8-s − 0.603·11-s − 1.10·13-s − 0.267·14-s + 1/4·16-s − 0.229·19-s − 0.426·22-s − 0.417·23-s − 25-s − 0.784·26-s − 0.188·28-s − 1.11·29-s − 0.718·31-s + 0.176·32-s − 0.986·37-s − 0.162·38-s + 0.937·41-s − 0.301·44-s − 0.294·46-s + 0.583·47-s + 1/7·49-s − 0.707·50-s − 0.554·52-s + 1.37·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 + T \) |
good | 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 6 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−8.552883285457483019151440390549, −7.39420774947384291991380176056, −7.27933586176350623461291056392, −5.96324113567297444862623324380, −5.53344493612271351602611461765, −4.53902655446390144799157235904, −3.75671759011358733432146880831, −2.74221403772722320427902199907, −1.90727701578356598386174083624, 0,
1.90727701578356598386174083624, 2.74221403772722320427902199907, 3.75671759011358733432146880831, 4.53902655446390144799157235904, 5.53344493612271351602611461765, 5.96324113567297444862623324380, 7.27933586176350623461291056392, 7.39420774947384291991380176056, 8.552883285457483019151440390549