| L(s) = 1 | − 3.07·3-s + 2.45·5-s − 4.52·7-s + 6.43·9-s + 0.159·11-s − 0.488·13-s − 7.54·15-s + 0.619·17-s − 2.21·19-s + 13.8·21-s + 1.05·23-s + 1.03·25-s − 10.5·27-s + 7.64·29-s + 5.50·31-s − 0.488·33-s − 11.1·35-s − 10.5·37-s + 1.50·39-s − 9.02·41-s − 1.71·43-s + 15.8·45-s + 0.546·47-s + 13.4·49-s − 1.90·51-s − 3.96·53-s + 0.391·55-s + ⋯ |
| L(s) = 1 | − 1.77·3-s + 1.09·5-s − 1.71·7-s + 2.14·9-s + 0.0479·11-s − 0.135·13-s − 1.94·15-s + 0.150·17-s − 0.508·19-s + 3.03·21-s + 0.220·23-s + 0.207·25-s − 2.02·27-s + 1.42·29-s + 0.988·31-s − 0.0850·33-s − 1.87·35-s − 1.73·37-s + 0.240·39-s − 1.40·41-s − 0.261·43-s + 2.35·45-s + 0.0797·47-s + 1.92·49-s − 0.266·51-s − 0.544·53-s + 0.0527·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 251 | \( 1 + T \) |
| good | 3 | \( 1 + 3.07T + 3T^{2} \) |
| 5 | \( 1 - 2.45T + 5T^{2} \) |
| 7 | \( 1 + 4.52T + 7T^{2} \) |
| 11 | \( 1 - 0.159T + 11T^{2} \) |
| 13 | \( 1 + 0.488T + 13T^{2} \) |
| 17 | \( 1 - 0.619T + 17T^{2} \) |
| 19 | \( 1 + 2.21T + 19T^{2} \) |
| 23 | \( 1 - 1.05T + 23T^{2} \) |
| 29 | \( 1 - 7.64T + 29T^{2} \) |
| 31 | \( 1 - 5.50T + 31T^{2} \) |
| 37 | \( 1 + 10.5T + 37T^{2} \) |
| 41 | \( 1 + 9.02T + 41T^{2} \) |
| 43 | \( 1 + 1.71T + 43T^{2} \) |
| 47 | \( 1 - 0.546T + 47T^{2} \) |
| 53 | \( 1 + 3.96T + 53T^{2} \) |
| 59 | \( 1 + 1.79T + 59T^{2} \) |
| 61 | \( 1 - 6.54T + 61T^{2} \) |
| 67 | \( 1 - 13.5T + 67T^{2} \) |
| 71 | \( 1 - 12.8T + 71T^{2} \) |
| 73 | \( 1 - 3.82T + 73T^{2} \) |
| 79 | \( 1 + 6.87T + 79T^{2} \) |
| 83 | \( 1 - 6.43T + 83T^{2} \) |
| 89 | \( 1 - 15.9T + 89T^{2} \) |
| 97 | \( 1 + 18.6T + 97T^{2} \) |
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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
−6.78128519533525753405326614234, −6.69489687818914403852829088783, −6.26210482266339232807689195332, −5.40821950443735064706700316781, −5.06631207559511902618515127227, −4.02501694662816961834289170782, −3.12512862769637889387557212446, −2.08347659837327868287429885021, −0.972578903374794260446186773844, 0,
0.972578903374794260446186773844, 2.08347659837327868287429885021, 3.12512862769637889387557212446, 4.02501694662816961834289170782, 5.06631207559511902618515127227, 5.40821950443735064706700316781, 6.26210482266339232807689195332, 6.69489687818914403852829088783, 6.78128519533525753405326614234
