| L(s) = 1 | − 2-s + 4-s + 4·7-s − 8-s − 4·11-s + 4·13-s − 4·14-s + 16-s + 4·19-s + 4·22-s + 4·23-s − 4·26-s + 4·28-s + 2·29-s − 4·31-s − 32-s + 8·37-s − 4·38-s − 8·41-s − 8·43-s − 4·44-s − 4·46-s − 8·47-s + 9·49-s + 4·52-s − 2·53-s − 4·56-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 1.51·7-s − 0.353·8-s − 1.20·11-s + 1.10·13-s − 1.06·14-s + 1/4·16-s + 0.917·19-s + 0.852·22-s + 0.834·23-s − 0.784·26-s + 0.755·28-s + 0.371·29-s − 0.718·31-s − 0.176·32-s + 1.31·37-s − 0.648·38-s − 1.24·41-s − 1.21·43-s − 0.603·44-s − 0.589·46-s − 1.16·47-s + 9/7·49-s + 0.554·52-s − 0.274·53-s − 0.534·56-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 167 | \( 1 - T \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−14.51698192461308, −13.72189355007936, −13.29554450913568, −12.98514970675436, −12.07848038499488, −11.66469414306792, −11.14157537464262, −10.95008114143272, −10.30507988921258, −9.848865093266387, −9.182446287193948, −8.566941680040937, −8.187147492796998, −7.899664905047044, −7.241017826583611, −6.766404291399753, −5.965350886219347, −5.379807274018103, −5.004381074712636, −4.397812186096918, −3.485895097796214, −2.971183875967944, −2.210960860824540, −1.481222394417088, −1.080007653601440, 0,
1.080007653601440, 1.481222394417088, 2.210960860824540, 2.971183875967944, 3.485895097796214, 4.397812186096918, 5.004381074712636, 5.379807274018103, 5.965350886219347, 6.766404291399753, 7.241017826583611, 7.899664905047044, 8.187147492796998, 8.566941680040937, 9.182446287193948, 9.848865093266387, 10.30507988921258, 10.95008114143272, 11.14157537464262, 11.66469414306792, 12.07848038499488, 12.98514970675436, 13.29554450913568, 13.72189355007936, 14.51698192461308
