| L(s) = 1 | + 3.05·3-s + 5-s + 1.28·7-s + 6.36·9-s + 3.58·11-s + 1.14·13-s + 3.05·15-s + 2.92·17-s + 2.71·19-s + 3.91·21-s + 23-s + 25-s + 10.2·27-s − 0.138·29-s + 0.305·31-s + 10.9·33-s + 1.28·35-s + 1.78·37-s + 3.49·39-s + 3.89·41-s − 3.33·43-s + 6.36·45-s − 5.69·47-s − 5.36·49-s + 8.93·51-s − 11.0·53-s + 3.58·55-s + ⋯ |
| L(s) = 1 | + 1.76·3-s + 0.447·5-s + 0.483·7-s + 2.12·9-s + 1.07·11-s + 0.316·13-s + 0.789·15-s + 0.708·17-s + 0.623·19-s + 0.854·21-s + 0.208·23-s + 0.200·25-s + 1.97·27-s − 0.0256·29-s + 0.0548·31-s + 1.90·33-s + 0.216·35-s + 0.292·37-s + 0.559·39-s + 0.608·41-s − 0.509·43-s + 0.948·45-s − 0.830·47-s − 0.765·49-s + 1.25·51-s − 1.51·53-s + 0.482·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.689522131\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.689522131\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 - 3.05T + 3T^{2} \) |
| 7 | \( 1 - 1.28T + 7T^{2} \) |
| 11 | \( 1 - 3.58T + 11T^{2} \) |
| 13 | \( 1 - 1.14T + 13T^{2} \) |
| 17 | \( 1 - 2.92T + 17T^{2} \) |
| 19 | \( 1 - 2.71T + 19T^{2} \) |
| 29 | \( 1 + 0.138T + 29T^{2} \) |
| 31 | \( 1 - 0.305T + 31T^{2} \) |
| 37 | \( 1 - 1.78T + 37T^{2} \) |
| 41 | \( 1 - 3.89T + 41T^{2} \) |
| 43 | \( 1 + 3.33T + 43T^{2} \) |
| 47 | \( 1 + 5.69T + 47T^{2} \) |
| 53 | \( 1 + 11.0T + 53T^{2} \) |
| 59 | \( 1 + 0.997T + 59T^{2} \) |
| 61 | \( 1 + 12.0T + 61T^{2} \) |
| 67 | \( 1 + 1.94T + 67T^{2} \) |
| 71 | \( 1 + 5.56T + 71T^{2} \) |
| 73 | \( 1 + 1.12T + 73T^{2} \) |
| 79 | \( 1 + 9.66T + 79T^{2} \) |
| 83 | \( 1 + 15.7T + 83T^{2} \) |
| 89 | \( 1 - 2.98T + 89T^{2} \) |
| 97 | \( 1 - 2.25T + 97T^{2} \) |
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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
−7.927536173008003367935888458879, −7.47998953434630538389375233869, −6.66061633726112396021098804964, −5.91321407091546510710036083249, −4.84962334948584219117458604015, −4.18207671346319931529408288275, −3.32505759818090663364607834582, −2.88134285596969179006321193928, −1.71127965218141738493376474085, −1.32902312684442831208194811213,
1.32902312684442831208194811213, 1.71127965218141738493376474085, 2.88134285596969179006321193928, 3.32505759818090663364607834582, 4.18207671346319931529408288275, 4.84962334948584219117458604015, 5.91321407091546510710036083249, 6.66061633726112396021098804964, 7.47998953434630538389375233869, 7.927536173008003367935888458879
