L(s) = 1 | − 2.55·3-s + 2.96·5-s − 0.820·7-s + 3.51·9-s + 3.75·11-s − 2.63·13-s − 7.56·15-s + 5.32·17-s + 5.04·19-s + 2.09·21-s − 2.33·23-s + 3.78·25-s − 1.31·27-s + 0.0387·29-s + 4.75·31-s − 9.58·33-s − 2.43·35-s − 1.34·37-s + 6.73·39-s + 0.422·41-s − 11.8·43-s + 10.4·45-s + 12.0·47-s − 6.32·49-s − 13.5·51-s + 2.90·53-s + 11.1·55-s + ⋯ |
L(s) = 1 | − 1.47·3-s + 1.32·5-s − 0.310·7-s + 1.17·9-s + 1.13·11-s − 0.731·13-s − 1.95·15-s + 1.29·17-s + 1.15·19-s + 0.456·21-s − 0.487·23-s + 0.757·25-s − 0.252·27-s + 0.00719·29-s + 0.853·31-s − 1.66·33-s − 0.411·35-s − 0.221·37-s + 1.07·39-s + 0.0659·41-s − 1.80·43-s + 1.55·45-s + 1.76·47-s − 0.903·49-s − 1.90·51-s + 0.398·53-s + 1.50·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.574168820\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.574168820\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.55T + 3T^{2} \) |
| 5 | \( 1 - 2.96T + 5T^{2} \) |
| 7 | \( 1 + 0.820T + 7T^{2} \) |
| 11 | \( 1 - 3.75T + 11T^{2} \) |
| 13 | \( 1 + 2.63T + 13T^{2} \) |
| 17 | \( 1 - 5.32T + 17T^{2} \) |
| 19 | \( 1 - 5.04T + 19T^{2} \) |
| 23 | \( 1 + 2.33T + 23T^{2} \) |
| 29 | \( 1 - 0.0387T + 29T^{2} \) |
| 31 | \( 1 - 4.75T + 31T^{2} \) |
| 37 | \( 1 + 1.34T + 37T^{2} \) |
| 41 | \( 1 - 0.422T + 41T^{2} \) |
| 43 | \( 1 + 11.8T + 43T^{2} \) |
| 47 | \( 1 - 12.0T + 47T^{2} \) |
| 53 | \( 1 - 2.90T + 53T^{2} \) |
| 59 | \( 1 - 13.8T + 59T^{2} \) |
| 61 | \( 1 - 1.76T + 61T^{2} \) |
| 67 | \( 1 + 3.46T + 67T^{2} \) |
| 71 | \( 1 - 6.11T + 71T^{2} \) |
| 73 | \( 1 - 2.52T + 73T^{2} \) |
| 79 | \( 1 + 0.164T + 79T^{2} \) |
| 83 | \( 1 + 14.9T + 83T^{2} \) |
| 89 | \( 1 + 13.3T + 89T^{2} \) |
| 97 | \( 1 - 4.30T + 97T^{2} \) |
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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
−8.546834072961169360838437297350, −7.40240121695098463017478235373, −6.72489665636114260609351510173, −6.14837416134540629930368224226, −5.48519668456024469597139177691, −5.10940061306206229946275857761, −3.99802004910107158420163397394, −2.89931760916935508985430989218, −1.66209411391971203035673562718, −0.827569532234503246548533699865,
0.827569532234503246548533699865, 1.66209411391971203035673562718, 2.89931760916935508985430989218, 3.99802004910107158420163397394, 5.10940061306206229946275857761, 5.48519668456024469597139177691, 6.14837416134540629930368224226, 6.72489665636114260609351510173, 7.40240121695098463017478235373, 8.546834072961169360838437297350