| L(s) = 1 | + 2·3-s + 4·5-s + 7-s + 9-s + 2·11-s − 4·13-s + 8·15-s + 2·17-s − 6·19-s + 2·21-s + 11·25-s − 4·27-s + 8·29-s + 8·31-s + 4·33-s + 4·35-s + 8·37-s − 8·39-s − 10·41-s + 2·43-s + 4·45-s − 8·47-s + 49-s + 4·51-s + 8·55-s − 12·57-s − 10·59-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 1.78·5-s + 0.377·7-s + 1/3·9-s + 0.603·11-s − 1.10·13-s + 2.06·15-s + 0.485·17-s − 1.37·19-s + 0.436·21-s + 11/5·25-s − 0.769·27-s + 1.48·29-s + 1.43·31-s + 0.696·33-s + 0.676·35-s + 1.31·37-s − 1.28·39-s − 1.56·41-s + 0.304·43-s + 0.596·45-s − 1.16·47-s + 1/7·49-s + 0.560·51-s + 1.07·55-s − 1.58·57-s − 1.30·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.617445250\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.617445250\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 5 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 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
−9.318761612207068811842344594127, −8.546625992271387674816885839022, −7.992914711579459891593754479777, −6.74776090668964381863923678873, −6.22556455636945214318257476470, −5.16624989138851774396165460424, −4.36535540120057539344813073391, −2.92567826122973580405986936408, −2.37367773065610120834468856295, −1.45832384904657446684298443518,
1.45832384904657446684298443518, 2.37367773065610120834468856295, 2.92567826122973580405986936408, 4.36535540120057539344813073391, 5.16624989138851774396165460424, 6.22556455636945214318257476470, 6.74776090668964381863923678873, 7.992914711579459891593754479777, 8.546625992271387674816885839022, 9.318761612207068811842344594127
