| L(s) = 1 | − 2.49·3-s − 2.70·5-s − 3.48·7-s + 3.20·9-s + 4.95·11-s − 1.54·13-s + 6.74·15-s + 1.95·17-s + 4.64·19-s + 8.67·21-s + 2.32·25-s − 0.516·27-s + 0.0794·29-s + 6.97·31-s − 12.3·33-s + 9.42·35-s − 11.5·37-s + 3.84·39-s + 8.96·41-s + 9.79·43-s − 8.67·45-s − 7.58·47-s + 5.12·49-s − 4.87·51-s − 9.11·53-s − 13.4·55-s − 11.5·57-s + ⋯ |
| L(s) = 1 | − 1.43·3-s − 1.21·5-s − 1.31·7-s + 1.06·9-s + 1.49·11-s − 0.428·13-s + 1.74·15-s + 0.474·17-s + 1.06·19-s + 1.89·21-s + 0.464·25-s − 0.0993·27-s + 0.0147·29-s + 1.25·31-s − 2.14·33-s + 1.59·35-s − 1.89·37-s + 0.615·39-s + 1.39·41-s + 1.49·43-s − 1.29·45-s − 1.10·47-s + 0.732·49-s − 0.682·51-s − 1.25·53-s − 1.80·55-s − 1.53·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8464 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8464 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5658986304\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5658986304\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 23 | \( 1 \) |
| good | 3 | \( 1 + 2.49T + 3T^{2} \) |
| 5 | \( 1 + 2.70T + 5T^{2} \) |
| 7 | \( 1 + 3.48T + 7T^{2} \) |
| 11 | \( 1 - 4.95T + 11T^{2} \) |
| 13 | \( 1 + 1.54T + 13T^{2} \) |
| 17 | \( 1 - 1.95T + 17T^{2} \) |
| 19 | \( 1 - 4.64T + 19T^{2} \) |
| 29 | \( 1 - 0.0794T + 29T^{2} \) |
| 31 | \( 1 - 6.97T + 31T^{2} \) |
| 37 | \( 1 + 11.5T + 37T^{2} \) |
| 41 | \( 1 - 8.96T + 41T^{2} \) |
| 43 | \( 1 - 9.79T + 43T^{2} \) |
| 47 | \( 1 + 7.58T + 47T^{2} \) |
| 53 | \( 1 + 9.11T + 53T^{2} \) |
| 59 | \( 1 + 8.19T + 59T^{2} \) |
| 61 | \( 1 - 8.12T + 61T^{2} \) |
| 67 | \( 1 - 3.24T + 67T^{2} \) |
| 71 | \( 1 - 4.39T + 71T^{2} \) |
| 73 | \( 1 + 14.8T + 73T^{2} \) |
| 79 | \( 1 + 5.71T + 79T^{2} \) |
| 83 | \( 1 - 0.464T + 83T^{2} \) |
| 89 | \( 1 - 1.93T + 89T^{2} \) |
| 97 | \( 1 - 1.69T + 97T^{2} \) |
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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
−7.47466000520438866533654431539, −7.02020997082578198194825556905, −6.35728362573121386812183881463, −5.90614452366703751971218516981, −5.00206234798551048692716231585, −4.28442045285917804788496756404, −3.60389870404110648726495768632, −2.95340743608295712681185326415, −1.27836200872643059360517844110, −0.44937497564043858549330913244,
0.44937497564043858549330913244, 1.27836200872643059360517844110, 2.95340743608295712681185326415, 3.60389870404110648726495768632, 4.28442045285917804788496756404, 5.00206234798551048692716231585, 5.90614452366703751971218516981, 6.35728362573121386812183881463, 7.02020997082578198194825556905, 7.47466000520438866533654431539
