| L(s) = 1 | + 5-s − 3.46·7-s + 1.26·11-s + 13-s + 3.46·17-s − 6.73·19-s − 2.19·23-s + 25-s − 1.46·29-s − 1.26·31-s − 3.46·35-s + 10.9·37-s + 4.53·41-s + 0.732·43-s + 4.53·47-s + 4.99·49-s − 4.53·53-s + 1.26·55-s + 12.1·59-s + 1.46·61-s + 65-s − 11.8·67-s + 5.66·71-s − 4.39·77-s + 9.46·79-s − 4.53·83-s + 3.46·85-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 1.30·7-s + 0.382·11-s + 0.277·13-s + 0.840·17-s − 1.54·19-s − 0.457·23-s + 0.200·25-s − 0.271·29-s − 0.227·31-s − 0.585·35-s + 1.79·37-s + 0.708·41-s + 0.111·43-s + 0.661·47-s + 0.714·49-s − 0.623·53-s + 0.170·55-s + 1.58·59-s + 0.187·61-s + 0.124·65-s − 1.44·67-s + 0.671·71-s − 0.500·77-s + 1.06·79-s − 0.497·83-s + 0.375·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.645327079\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.645327079\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 3.46T + 7T^{2} \) |
| 11 | \( 1 - 1.26T + 11T^{2} \) |
| 17 | \( 1 - 3.46T + 17T^{2} \) |
| 19 | \( 1 + 6.73T + 19T^{2} \) |
| 23 | \( 1 + 2.19T + 23T^{2} \) |
| 29 | \( 1 + 1.46T + 29T^{2} \) |
| 31 | \( 1 + 1.26T + 31T^{2} \) |
| 37 | \( 1 - 10.9T + 37T^{2} \) |
| 41 | \( 1 - 4.53T + 41T^{2} \) |
| 43 | \( 1 - 0.732T + 43T^{2} \) |
| 47 | \( 1 - 4.53T + 47T^{2} \) |
| 53 | \( 1 + 4.53T + 53T^{2} \) |
| 59 | \( 1 - 12.1T + 59T^{2} \) |
| 61 | \( 1 - 1.46T + 61T^{2} \) |
| 67 | \( 1 + 11.8T + 67T^{2} \) |
| 71 | \( 1 - 5.66T + 71T^{2} \) |
| 73 | \( 1 + 73T^{2} \) |
| 79 | \( 1 - 9.46T + 79T^{2} \) |
| 83 | \( 1 + 4.53T + 83T^{2} \) |
| 89 | \( 1 + 8.92T + 89T^{2} \) |
| 97 | \( 1 - 15.8T + 97T^{2} \) |
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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
−8.355608977841201886229213353855, −7.54843955889632322470409194871, −6.67420983925517988369639976567, −6.14155407028209629546697711662, −5.65449622545179560378732811265, −4.43426261704847649746019308940, −3.76082560438513155618703777034, −2.89162085793543528291882639748, −2.01228439734809467432266652097, −0.69819855859719626857583175622,
0.69819855859719626857583175622, 2.01228439734809467432266652097, 2.89162085793543528291882639748, 3.76082560438513155618703777034, 4.43426261704847649746019308940, 5.65449622545179560378732811265, 6.14155407028209629546697711662, 6.67420983925517988369639976567, 7.54843955889632322470409194871, 8.355608977841201886229213353855
