| L(s) = 1 | − 2·3-s + 2·5-s + 9-s + 4·11-s + 6·13-s − 4·15-s + 4·17-s − 6·19-s + 4·23-s − 25-s + 4·27-s + 6·29-s − 4·31-s − 8·33-s + 6·37-s − 12·39-s − 4·41-s + 12·43-s + 2·45-s − 12·47-s − 8·51-s − 6·53-s + 8·55-s + 12·57-s − 6·59-s − 6·61-s + 12·65-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.894·5-s + 1/3·9-s + 1.20·11-s + 1.66·13-s − 1.03·15-s + 0.970·17-s − 1.37·19-s + 0.834·23-s − 1/5·25-s + 0.769·27-s + 1.11·29-s − 0.718·31-s − 1.39·33-s + 0.986·37-s − 1.92·39-s − 0.624·41-s + 1.82·43-s + 0.298·45-s − 1.75·47-s − 1.12·51-s − 0.824·53-s + 1.07·55-s + 1.58·57-s − 0.781·59-s − 0.768·61-s + 1.48·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3136 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.724331369\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.724331369\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 16 T + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
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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
−8.819905968577035763530381205853, −8.007570528889646360182709742364, −6.77337982382803703153854326408, −6.13353461788814750400884298266, −6.01668664610688358486037400226, −4.98059997077380522001525482948, −4.11425275026060409882723877727, −3.13759465813828745715366113036, −1.71459972849340553871017736531, −0.914109485961916639220233133899,
0.914109485961916639220233133899, 1.71459972849340553871017736531, 3.13759465813828745715366113036, 4.11425275026060409882723877727, 4.98059997077380522001525482948, 6.01668664610688358486037400226, 6.13353461788814750400884298266, 6.77337982382803703153854326408, 8.007570528889646360182709742364, 8.819905968577035763530381205853
