| L(s) = 1 | + 3-s − 5-s − 1.41·7-s + 9-s + 0.585·11-s + 0.585·13-s − 15-s − 2.82·17-s − 19-s − 1.41·21-s + 4.82·23-s + 25-s + 27-s − 7.07·29-s + 4.82·31-s + 0.585·33-s + 1.41·35-s + 6.24·37-s + 0.585·39-s + 9.89·41-s − 11.0·43-s − 45-s + 3.17·47-s − 5·49-s − 2.82·51-s + 8.48·53-s − 0.585·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s − 0.534·7-s + 0.333·9-s + 0.176·11-s + 0.162·13-s − 0.258·15-s − 0.685·17-s − 0.229·19-s − 0.308·21-s + 1.00·23-s + 0.200·25-s + 0.192·27-s − 1.31·29-s + 0.867·31-s + 0.101·33-s + 0.239·35-s + 1.02·37-s + 0.0938·39-s + 1.54·41-s − 1.68·43-s − 0.149·45-s + 0.462·47-s − 0.714·49-s − 0.396·51-s + 1.16·53-s − 0.0789·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.954076480\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.954076480\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| good | 7 | \( 1 + 1.41T + 7T^{2} \) |
| 11 | \( 1 - 0.585T + 11T^{2} \) |
| 13 | \( 1 - 0.585T + 13T^{2} \) |
| 17 | \( 1 + 2.82T + 17T^{2} \) |
| 23 | \( 1 - 4.82T + 23T^{2} \) |
| 29 | \( 1 + 7.07T + 29T^{2} \) |
| 31 | \( 1 - 4.82T + 31T^{2} \) |
| 37 | \( 1 - 6.24T + 37T^{2} \) |
| 41 | \( 1 - 9.89T + 41T^{2} \) |
| 43 | \( 1 + 11.0T + 43T^{2} \) |
| 47 | \( 1 - 3.17T + 47T^{2} \) |
| 53 | \( 1 - 8.48T + 53T^{2} \) |
| 59 | \( 1 - 1.17T + 59T^{2} \) |
| 61 | \( 1 + 1.65T + 61T^{2} \) |
| 67 | \( 1 - 11.3T + 67T^{2} \) |
| 71 | \( 1 - 14.8T + 71T^{2} \) |
| 73 | \( 1 + 11.6T + 73T^{2} \) |
| 79 | \( 1 - 13.6T + 79T^{2} \) |
| 83 | \( 1 - 7.65T + 83T^{2} \) |
| 89 | \( 1 + 12.2T + 89T^{2} \) |
| 97 | \( 1 - 11.8T + 97T^{2} \) |
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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
−8.313621582073231247171322693742, −7.68108958140523198463225719684, −6.87873310912038523048087510399, −6.35428729673210143120297189037, −5.32942091958046848645324615427, −4.42220171777366695328540447564, −3.74318430994777737947033940341, −2.95382513828096291069376149452, −2.07791088339101593293641021023, −0.75048504311458430834985929481,
0.75048504311458430834985929481, 2.07791088339101593293641021023, 2.95382513828096291069376149452, 3.74318430994777737947033940341, 4.42220171777366695328540447564, 5.32942091958046848645324615427, 6.35428729673210143120297189037, 6.87873310912038523048087510399, 7.68108958140523198463225719684, 8.313621582073231247171322693742
