| L(s) = 1 | − 5-s − 3.74·7-s + 2.54·11-s + 13-s + 1.74·17-s + 2.29·19-s − 1.74·23-s + 25-s − 4.29·29-s − 2·31-s + 3.74·35-s + 8.03·37-s − 2.94·41-s − 7.49·43-s − 3.49·47-s + 7.03·49-s + 2.54·53-s − 2.54·55-s − 8.58·59-s − 4.03·61-s − 65-s + 1.49·67-s + 13.5·71-s + 9.78·73-s − 9.52·77-s − 8.94·79-s − 1.74·85-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.41·7-s + 0.766·11-s + 0.277·13-s + 0.423·17-s + 0.525·19-s − 0.364·23-s + 0.200·25-s − 0.796·29-s − 0.359·31-s + 0.633·35-s + 1.32·37-s − 0.460·41-s − 1.14·43-s − 0.509·47-s + 1.00·49-s + 0.349·53-s − 0.342·55-s − 1.11·59-s − 0.516·61-s − 0.124·65-s + 0.182·67-s + 1.60·71-s + 1.14·73-s − 1.08·77-s − 1.00·79-s − 0.189·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.285666689\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.285666689\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 + 3.74T + 7T^{2} \) |
| 11 | \( 1 - 2.54T + 11T^{2} \) |
| 17 | \( 1 - 1.74T + 17T^{2} \) |
| 19 | \( 1 - 2.29T + 19T^{2} \) |
| 23 | \( 1 + 1.74T + 23T^{2} \) |
| 29 | \( 1 + 4.29T + 29T^{2} \) |
| 31 | \( 1 + 2T + 31T^{2} \) |
| 37 | \( 1 - 8.03T + 37T^{2} \) |
| 41 | \( 1 + 2.94T + 41T^{2} \) |
| 43 | \( 1 + 7.49T + 43T^{2} \) |
| 47 | \( 1 + 3.49T + 47T^{2} \) |
| 53 | \( 1 - 2.54T + 53T^{2} \) |
| 59 | \( 1 + 8.58T + 59T^{2} \) |
| 61 | \( 1 + 4.03T + 61T^{2} \) |
| 67 | \( 1 - 1.49T + 67T^{2} \) |
| 71 | \( 1 - 13.5T + 71T^{2} \) |
| 73 | \( 1 - 9.78T + 73T^{2} \) |
| 79 | \( 1 + 8.94T + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 - 6.94T + 89T^{2} \) |
| 97 | \( 1 - 5.34T + 97T^{2} \) |
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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
−7.68633791465993524879291635368, −6.92519698279130362148511753353, −6.40549509261351404250204180970, −5.80780671944891516855563399260, −4.94861201263458113288864005129, −3.95209389953810301789984434608, −3.51810235608278173706845653754, −2.84025493615072056279593454631, −1.65755062950661752796563538696, −0.54580667901964962008519825661,
0.54580667901964962008519825661, 1.65755062950661752796563538696, 2.84025493615072056279593454631, 3.51810235608278173706845653754, 3.95209389953810301789984434608, 4.94861201263458113288864005129, 5.80780671944891516855563399260, 6.40549509261351404250204180970, 6.92519698279130362148511753353, 7.68633791465993524879291635368
