| L(s) = 1 | + 3-s − 0.920·5-s − 4.87·7-s + 9-s − 0.964·11-s − 0.920·15-s + 2.47·17-s + 3.12·19-s − 4.87·21-s − 4.92·23-s − 4.15·25-s + 27-s + 6.90·29-s − 4.61·31-s − 0.964·33-s + 4.48·35-s + 4.81·37-s − 3.43·41-s + 10.0·43-s − 0.920·45-s − 8.64·47-s + 16.7·49-s + 2.47·51-s + 0.841·53-s + 0.887·55-s + 3.12·57-s − 9.35·59-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.411·5-s − 1.84·7-s + 0.333·9-s − 0.290·11-s − 0.237·15-s + 0.600·17-s + 0.717·19-s − 1.06·21-s − 1.02·23-s − 0.830·25-s + 0.192·27-s + 1.28·29-s − 0.828·31-s − 0.167·33-s + 0.757·35-s + 0.791·37-s − 0.536·41-s + 1.53·43-s − 0.137·45-s − 1.26·47-s + 2.38·49-s + 0.346·51-s + 0.115·53-s + 0.119·55-s + 0.414·57-s − 1.21·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.400886657\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.400886657\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 + 0.920T + 5T^{2} \) |
| 7 | \( 1 + 4.87T + 7T^{2} \) |
| 11 | \( 1 + 0.964T + 11T^{2} \) |
| 17 | \( 1 - 2.47T + 17T^{2} \) |
| 19 | \( 1 - 3.12T + 19T^{2} \) |
| 23 | \( 1 + 4.92T + 23T^{2} \) |
| 29 | \( 1 - 6.90T + 29T^{2} \) |
| 31 | \( 1 + 4.61T + 31T^{2} \) |
| 37 | \( 1 - 4.81T + 37T^{2} \) |
| 41 | \( 1 + 3.43T + 41T^{2} \) |
| 43 | \( 1 - 10.0T + 43T^{2} \) |
| 47 | \( 1 + 8.64T + 47T^{2} \) |
| 53 | \( 1 - 0.841T + 53T^{2} \) |
| 59 | \( 1 + 9.35T + 59T^{2} \) |
| 61 | \( 1 - 0.793T + 61T^{2} \) |
| 67 | \( 1 - 6.99T + 67T^{2} \) |
| 71 | \( 1 - 15.5T + 71T^{2} \) |
| 73 | \( 1 - 5.72T + 73T^{2} \) |
| 79 | \( 1 + 13.8T + 79T^{2} \) |
| 83 | \( 1 - 14.8T + 83T^{2} \) |
| 89 | \( 1 + 10.6T + 89T^{2} \) |
| 97 | \( 1 - 9.53T + 97T^{2} \) |
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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
−8.348946812652769155129453734678, −7.73820274982913587571653978328, −7.04717376164131821038053437856, −6.25891686694944377113966522231, −5.64507862523220335698992564278, −4.46677183737739336281727081792, −3.56170915418439695917379750530, −3.16529077725331483658634918040, −2.18685624984650365273124178163, −0.63135110701700118266770776544,
0.63135110701700118266770776544, 2.18685624984650365273124178163, 3.16529077725331483658634918040, 3.56170915418439695917379750530, 4.46677183737739336281727081792, 5.64507862523220335698992564278, 6.25891686694944377113966522231, 7.04717376164131821038053437856, 7.73820274982913587571653978328, 8.348946812652769155129453734678
