L(s) = 1 | + 1.85·3-s + 2.88·5-s − 1.11·7-s + 0.447·9-s + 4.87·11-s + 3.77·13-s + 5.36·15-s − 0.0587·17-s + 2.60·19-s − 2.06·21-s + 2.33·23-s + 3.34·25-s − 4.73·27-s + 6.17·29-s + 4.36·31-s + 9.05·33-s − 3.21·35-s − 4.47·37-s + 7.01·39-s − 2.97·41-s + 10.9·43-s + 1.29·45-s − 8.04·47-s − 5.76·49-s − 0.109·51-s − 11.7·53-s + 14.0·55-s + ⋯ |
L(s) = 1 | + 1.07·3-s + 1.29·5-s − 0.420·7-s + 0.149·9-s + 1.47·11-s + 1.04·13-s + 1.38·15-s − 0.0142·17-s + 0.596·19-s − 0.450·21-s + 0.486·23-s + 0.669·25-s − 0.912·27-s + 1.14·29-s + 0.784·31-s + 1.57·33-s − 0.543·35-s − 0.735·37-s + 1.12·39-s − 0.464·41-s + 1.66·43-s + 0.192·45-s − 1.17·47-s − 0.823·49-s − 0.0152·51-s − 1.60·53-s + 1.90·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.637420301\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.637420301\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 503 | \( 1 - T \) |
good | 3 | \( 1 - 1.85T + 3T^{2} \) |
| 5 | \( 1 - 2.88T + 5T^{2} \) |
| 7 | \( 1 + 1.11T + 7T^{2} \) |
| 11 | \( 1 - 4.87T + 11T^{2} \) |
| 13 | \( 1 - 3.77T + 13T^{2} \) |
| 17 | \( 1 + 0.0587T + 17T^{2} \) |
| 19 | \( 1 - 2.60T + 19T^{2} \) |
| 23 | \( 1 - 2.33T + 23T^{2} \) |
| 29 | \( 1 - 6.17T + 29T^{2} \) |
| 31 | \( 1 - 4.36T + 31T^{2} \) |
| 37 | \( 1 + 4.47T + 37T^{2} \) |
| 41 | \( 1 + 2.97T + 41T^{2} \) |
| 43 | \( 1 - 10.9T + 43T^{2} \) |
| 47 | \( 1 + 8.04T + 47T^{2} \) |
| 53 | \( 1 + 11.7T + 53T^{2} \) |
| 59 | \( 1 - 1.31T + 59T^{2} \) |
| 61 | \( 1 - 7.26T + 61T^{2} \) |
| 67 | \( 1 - 5.79T + 67T^{2} \) |
| 71 | \( 1 - 0.407T + 71T^{2} \) |
| 73 | \( 1 - 1.44T + 73T^{2} \) |
| 79 | \( 1 + 9.65T + 79T^{2} \) |
| 83 | \( 1 - 13.7T + 83T^{2} \) |
| 89 | \( 1 + 16.0T + 89T^{2} \) |
| 97 | \( 1 + 4.02T + 97T^{2} \) |
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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
−8.097299546735878133638598847107, −6.97601755023333886579371291836, −6.43203014066648237814202867887, −5.95863284770633896350026147492, −5.07456982892870620787526250365, −4.07095903865295253289773388920, −3.33940330495128956277511723269, −2.76129545910728488365386264492, −1.77777378314501831459649112197, −1.11915244386668743520572601330,
1.11915244386668743520572601330, 1.77777378314501831459649112197, 2.76129545910728488365386264492, 3.33940330495128956277511723269, 4.07095903865295253289773388920, 5.07456982892870620787526250365, 5.95863284770633896350026147492, 6.43203014066648237814202867887, 6.97601755023333886579371291836, 8.097299546735878133638598847107