| L(s) = 1 | + 2-s + 4-s + 4·5-s − 3·7-s + 8-s + 4·10-s − 2·11-s + 7·13-s − 3·14-s + 16-s + 4·20-s − 2·22-s + 4·23-s + 11·25-s + 7·26-s − 3·28-s + 4·29-s − 31-s + 32-s − 12·35-s − 7·37-s + 4·40-s + 4·41-s + 7·43-s − 2·44-s + 4·46-s − 2·47-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 1.78·5-s − 1.13·7-s + 0.353·8-s + 1.26·10-s − 0.603·11-s + 1.94·13-s − 0.801·14-s + 1/4·16-s + 0.894·20-s − 0.426·22-s + 0.834·23-s + 11/5·25-s + 1.37·26-s − 0.566·28-s + 0.742·29-s − 0.179·31-s + 0.176·32-s − 2.02·35-s − 1.15·37-s + 0.632·40-s + 0.624·41-s + 1.06·43-s − 0.301·44-s + 0.589·46-s − 0.291·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6498 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6498 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.570582273\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.570582273\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 19 | \( 1 \) |
| good | 5 | \( 1 - 4 T + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 7 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 - 7 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 + 3 T + p T^{2} \) |
| 79 | \( 1 + 5 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−7.993941571045357984848329694719, −6.84242165298359965046149067902, −6.45764666986659842858926628131, −5.85765191651095015789782715737, −5.42599672749247176261061258986, −4.48338128396501106920044648197, −3.37594549239590243314015560198, −2.92399732618539755908664351737, −1.96329651407266469694136564224, −1.05053555387539568764665621009,
1.05053555387539568764665621009, 1.96329651407266469694136564224, 2.92399732618539755908664351737, 3.37594549239590243314015560198, 4.48338128396501106920044648197, 5.42599672749247176261061258986, 5.85765191651095015789782715737, 6.45764666986659842858926628131, 6.84242165298359965046149067902, 7.993941571045357984848329694719
