| L(s) = 1 | − 3-s − 5-s + 2.70·7-s + 9-s + 0.701·11-s − 13-s + 15-s − 0.701·17-s − 2·19-s − 2.70·21-s + 6.70·23-s + 25-s − 27-s + 9.40·29-s − 9.40·31-s − 0.701·33-s − 2.70·35-s − 6.70·37-s + 39-s + 10.7·41-s + 4·43-s − 45-s − 1.40·47-s + 0.298·49-s + 0.701·51-s + 10.7·53-s − 0.701·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 1.02·7-s + 0.333·9-s + 0.211·11-s − 0.277·13-s + 0.258·15-s − 0.170·17-s − 0.458·19-s − 0.589·21-s + 1.39·23-s + 0.200·25-s − 0.192·27-s + 1.74·29-s − 1.68·31-s − 0.122·33-s − 0.456·35-s − 1.10·37-s + 0.160·39-s + 1.67·41-s + 0.609·43-s − 0.149·45-s − 0.204·47-s + 0.0426·49-s + 0.0982·51-s + 1.46·53-s − 0.0945·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.425110405\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.425110405\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 - 2.70T + 7T^{2} \) |
| 11 | \( 1 - 0.701T + 11T^{2} \) |
| 17 | \( 1 + 0.701T + 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 23 | \( 1 - 6.70T + 23T^{2} \) |
| 29 | \( 1 - 9.40T + 29T^{2} \) |
| 31 | \( 1 + 9.40T + 31T^{2} \) |
| 37 | \( 1 + 6.70T + 37T^{2} \) |
| 41 | \( 1 - 10.7T + 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 + 1.40T + 47T^{2} \) |
| 53 | \( 1 - 10.7T + 53T^{2} \) |
| 59 | \( 1 + 14.8T + 59T^{2} \) |
| 61 | \( 1 - 2.70T + 61T^{2} \) |
| 67 | \( 1 + 4T + 67T^{2} \) |
| 71 | \( 1 - 15.5T + 71T^{2} \) |
| 73 | \( 1 - 13.4T + 73T^{2} \) |
| 79 | \( 1 - 4.70T + 79T^{2} \) |
| 83 | \( 1 - 4T + 83T^{2} \) |
| 89 | \( 1 - 8.10T + 89T^{2} \) |
| 97 | \( 1 - 18.1T + 97T^{2} \) |
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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
−9.307358576304214997750921444045, −8.666376700763651782389743663831, −7.74419814236012157091093863671, −7.09378670714638701836806674232, −6.20554913163162826846281664299, −5.10559675455828148694822898327, −4.63630331963000520860375565331, −3.56362766920041246015856934915, −2.19599287041094763525578487798, −0.894035209469996076943091148765,
0.894035209469996076943091148765, 2.19599287041094763525578487798, 3.56362766920041246015856934915, 4.63630331963000520860375565331, 5.10559675455828148694822898327, 6.20554913163162826846281664299, 7.09378670714638701836806674232, 7.74419814236012157091093863671, 8.666376700763651782389743663831, 9.307358576304214997750921444045
