| L(s) = 1 | + 3-s + 5-s + 2.78·7-s + 9-s + 5.21·11-s − 0.429·13-s + 15-s + 2.09·17-s − 1.64·19-s + 2.78·21-s − 1.54·23-s + 25-s + 27-s − 9.19·29-s + 31-s + 5.21·33-s + 2.78·35-s − 2.42·37-s − 0.429·39-s + 8.66·41-s + 8.50·43-s + 45-s − 4.95·47-s + 0.744·49-s + 2.09·51-s + 12.2·53-s + 5.21·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s + 1.05·7-s + 0.333·9-s + 1.57·11-s − 0.119·13-s + 0.258·15-s + 0.508·17-s − 0.377·19-s + 0.607·21-s − 0.322·23-s + 0.200·25-s + 0.192·27-s − 1.70·29-s + 0.179·31-s + 0.907·33-s + 0.470·35-s − 0.399·37-s − 0.0687·39-s + 1.35·41-s + 1.29·43-s + 0.149·45-s − 0.723·47-s + 0.106·49-s + 0.293·51-s + 1.67·53-s + 0.702·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.810973299\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.810973299\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 31 | \( 1 - T \) |
| good | 7 | \( 1 - 2.78T + 7T^{2} \) |
| 11 | \( 1 - 5.21T + 11T^{2} \) |
| 13 | \( 1 + 0.429T + 13T^{2} \) |
| 17 | \( 1 - 2.09T + 17T^{2} \) |
| 19 | \( 1 + 1.64T + 19T^{2} \) |
| 23 | \( 1 + 1.54T + 23T^{2} \) |
| 29 | \( 1 + 9.19T + 29T^{2} \) |
| 37 | \( 1 + 2.42T + 37T^{2} \) |
| 41 | \( 1 - 8.66T + 41T^{2} \) |
| 43 | \( 1 - 8.50T + 43T^{2} \) |
| 47 | \( 1 + 4.95T + 47T^{2} \) |
| 53 | \( 1 - 12.2T + 53T^{2} \) |
| 59 | \( 1 + 2.33T + 59T^{2} \) |
| 61 | \( 1 - 11.7T + 61T^{2} \) |
| 67 | \( 1 - 5.72T + 67T^{2} \) |
| 71 | \( 1 - 9.07T + 71T^{2} \) |
| 73 | \( 1 - 1.68T + 73T^{2} \) |
| 79 | \( 1 + 10.8T + 79T^{2} \) |
| 83 | \( 1 - 2.99T + 83T^{2} \) |
| 89 | \( 1 - 2.21T + 89T^{2} \) |
| 97 | \( 1 - 11.7T + 97T^{2} \) |
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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
−7.919001320582276528501725322160, −7.28972965959908670474128361564, −6.56874060858090548039812010266, −5.77311169975145517112477719627, −5.10252659909510693796327093686, −4.07446507061839358857193077352, −3.79565110362366607696997039944, −2.51835581338761973358941057370, −1.80580433469136978331845073867, −1.04305229093114937405595290748,
1.04305229093114937405595290748, 1.80580433469136978331845073867, 2.51835581338761973358941057370, 3.79565110362366607696997039944, 4.07446507061839358857193077352, 5.10252659909510693796327093686, 5.77311169975145517112477719627, 6.56874060858090548039812010266, 7.28972965959908670474128361564, 7.919001320582276528501725322160
