| L(s) = 1 | − 5-s − 7-s + 2·13-s − 4·17-s + 2·19-s + 25-s + 2·29-s − 8·31-s + 35-s − 10·37-s + 6·41-s + 4·43-s + 12·47-s + 49-s − 2·53-s + 10·59-s + 2·61-s − 2·65-s + 2·67-s − 4·71-s + 6·73-s + 4·79-s + 4·83-s + 4·85-s + 10·89-s − 2·91-s − 2·95-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.377·7-s + 0.554·13-s − 0.970·17-s + 0.458·19-s + 1/5·25-s + 0.371·29-s − 1.43·31-s + 0.169·35-s − 1.64·37-s + 0.937·41-s + 0.609·43-s + 1.75·47-s + 1/7·49-s − 0.274·53-s + 1.30·59-s + 0.256·61-s − 0.248·65-s + 0.244·67-s − 0.474·71-s + 0.702·73-s + 0.450·79-s + 0.439·83-s + 0.433·85-s + 1.05·89-s − 0.209·91-s − 0.205·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 152460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 152460 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.742314534\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.742314534\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 4 T + p T^{2} \) |
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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
−13.37039755768073, −12.80761420358047, −12.36756689740317, −11.99702881171160, −11.25486006106080, −11.07822650884932, −10.46347911130615, −10.11457311357493, −9.245280427274056, −9.045617327855572, −8.616230058155048, −7.959870900558350, −7.386677201694671, −7.050765038449334, −6.479795096082811, −5.928247377042882, −5.387432768931399, −4.864522755697614, −4.130181339079106, −3.745056116004589, −3.265030515063446, −2.450619783780116, −1.997774769170686, −1.078665977128429, −0.4319633931324262,
0.4319633931324262, 1.078665977128429, 1.997774769170686, 2.450619783780116, 3.265030515063446, 3.745056116004589, 4.130181339079106, 4.864522755697614, 5.387432768931399, 5.928247377042882, 6.479795096082811, 7.050765038449334, 7.386677201694671, 7.959870900558350, 8.616230058155048, 9.045617327855572, 9.245280427274056, 10.11457311357493, 10.46347911130615, 11.07822650884932, 11.25486006106080, 11.99702881171160, 12.36756689740317, 12.80761420358047, 13.37039755768073
