| L(s) = 1 | + 3-s − 5-s − 7-s − 2·9-s + 3·11-s − 15-s − 3·17-s + 5·19-s − 21-s + 9·23-s + 25-s − 5·27-s − 9·29-s + 8·31-s + 3·33-s + 35-s − 7·37-s + 3·41-s − 43-s + 2·45-s − 6·49-s − 3·51-s + 6·53-s − 3·55-s + 5·57-s + 9·59-s − 61-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s − 0.377·7-s − 2/3·9-s + 0.904·11-s − 0.258·15-s − 0.727·17-s + 1.14·19-s − 0.218·21-s + 1.87·23-s + 1/5·25-s − 0.962·27-s − 1.67·29-s + 1.43·31-s + 0.522·33-s + 0.169·35-s − 1.15·37-s + 0.468·41-s − 0.152·43-s + 0.298·45-s − 6/7·49-s − 0.420·51-s + 0.824·53-s − 0.404·55-s + 0.662·57-s + 1.17·59-s − 0.128·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.926485537\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.926485537\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 - 9 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 - 9 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 3 T + p T^{2} \) |
| 97 | \( 1 - 17 T + p T^{2} \) |
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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
−8.774254613940752774407048566813, −7.916476304208814150893106411156, −7.12822152433978872301932794634, −6.55093204068496153605834035820, −5.54705035786467537425153216397, −4.76021950214809217589221275910, −3.64105965140846546198503519477, −3.22099085877514272314377655347, −2.16579952955666613960322268203, −0.801352727206773013022012262069,
0.801352727206773013022012262069, 2.16579952955666613960322268203, 3.22099085877514272314377655347, 3.64105965140846546198503519477, 4.76021950214809217589221275910, 5.54705035786467537425153216397, 6.55093204068496153605834035820, 7.12822152433978872301932794634, 7.916476304208814150893106411156, 8.774254613940752774407048566813
