| L(s) = 1 | + 1.56·3-s + 3.56·5-s − 0.561·9-s − 11-s + 5.12·13-s + 5.56·15-s + 2·17-s + 3.12·19-s + 5.56·23-s + 7.68·25-s − 5.56·27-s − 2·29-s − 6.43·31-s − 1.56·33-s + 0.438·37-s + 8·39-s + 10·41-s − 4·43-s − 2·45-s + 10.2·47-s + 3.12·51-s + 12.2·53-s − 3.56·55-s + 4.87·57-s − 9.56·59-s − 12.2·61-s + 18.2·65-s + ⋯ |
| L(s) = 1 | + 0.901·3-s + 1.59·5-s − 0.187·9-s − 0.301·11-s + 1.42·13-s + 1.43·15-s + 0.485·17-s + 0.716·19-s + 1.15·23-s + 1.53·25-s − 1.07·27-s − 0.371·29-s − 1.15·31-s − 0.271·33-s + 0.0720·37-s + 1.28·39-s + 1.56·41-s − 0.609·43-s − 0.298·45-s + 1.49·47-s + 0.437·51-s + 1.68·53-s − 0.480·55-s + 0.645·57-s − 1.24·59-s − 1.56·61-s + 2.26·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.490369998\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.490369998\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 3 | \( 1 - 1.56T + 3T^{2} \) |
| 5 | \( 1 - 3.56T + 5T^{2} \) |
| 13 | \( 1 - 5.12T + 13T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 19 | \( 1 - 3.12T + 19T^{2} \) |
| 23 | \( 1 - 5.56T + 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 + 6.43T + 31T^{2} \) |
| 37 | \( 1 - 0.438T + 37T^{2} \) |
| 41 | \( 1 - 10T + 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 - 10.2T + 47T^{2} \) |
| 53 | \( 1 - 12.2T + 53T^{2} \) |
| 59 | \( 1 + 9.56T + 59T^{2} \) |
| 61 | \( 1 + 12.2T + 61T^{2} \) |
| 67 | \( 1 + 1.56T + 67T^{2} \) |
| 71 | \( 1 - 8.68T + 71T^{2} \) |
| 73 | \( 1 + 12.2T + 73T^{2} \) |
| 79 | \( 1 - 3.12T + 79T^{2} \) |
| 83 | \( 1 + 8T + 83T^{2} \) |
| 89 | \( 1 - 8.43T + 89T^{2} \) |
| 97 | \( 1 + 4.43T + 97T^{2} \) |
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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
−7.77338903999988000038602703280, −7.20494157532337757555080033874, −6.21807935631760219587492575700, −5.70168659769389723638079340182, −5.28275303428965737912242683834, −4.08815196912956055196668401363, −3.23281970372005801767444422986, −2.69208749093085038624165891354, −1.83063504038518845200777674797, −1.06271186926021691802176623021,
1.06271186926021691802176623021, 1.83063504038518845200777674797, 2.69208749093085038624165891354, 3.23281970372005801767444422986, 4.08815196912956055196668401363, 5.28275303428965737912242683834, 5.70168659769389723638079340182, 6.21807935631760219587492575700, 7.20494157532337757555080033874, 7.77338903999988000038602703280
