| L(s) = 1 | − 3-s − 5-s − 2·7-s + 9-s − 3·11-s + 2·13-s + 15-s − 2·19-s + 2·21-s − 3·23-s + 25-s − 27-s − 29-s − 8·31-s + 3·33-s + 2·35-s − 37-s − 2·39-s − 3·41-s + 43-s − 45-s + 6·47-s − 3·49-s − 3·53-s + 3·55-s + 2·57-s + 12·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 0.755·7-s + 1/3·9-s − 0.904·11-s + 0.554·13-s + 0.258·15-s − 0.458·19-s + 0.436·21-s − 0.625·23-s + 1/5·25-s − 0.192·27-s − 0.185·29-s − 1.43·31-s + 0.522·33-s + 0.338·35-s − 0.164·37-s − 0.320·39-s − 0.468·41-s + 0.152·43-s − 0.149·45-s + 0.875·47-s − 3/7·49-s − 0.412·53-s + 0.404·55-s + 0.264·57-s + 1.56·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6495700638\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6495700638\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 29 | \( 1 + T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 14 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 11 T + p T^{2} \) |
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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
−7.899770801579861473981087804378, −7.17531900377998881845205753889, −6.58180755376957205188814173616, −5.77006488668393121681534499142, −5.30160500230118692777924738924, −4.27779225259182179557770504169, −3.67756057504434039559122088317, −2.80266064625643525252358409624, −1.75869855474454202455008515450, −0.41094424430962889401150007147,
0.41094424430962889401150007147, 1.75869855474454202455008515450, 2.80266064625643525252358409624, 3.67756057504434039559122088317, 4.27779225259182179557770504169, 5.30160500230118692777924738924, 5.77006488668393121681534499142, 6.58180755376957205188814173616, 7.17531900377998881845205753889, 7.899770801579861473981087804378
