| L(s) = 1 | + 1.62·3-s + 1.90·5-s − 7-s − 0.354·9-s + 11-s − 13-s + 3.10·15-s − 3.20·17-s − 1.68·19-s − 1.62·21-s − 6.28·23-s − 1.35·25-s − 5.45·27-s − 10.4·29-s + 10.9·31-s + 1.62·33-s − 1.90·35-s + 7.50·37-s − 1.62·39-s + 4.62·41-s + 6.59·43-s − 0.676·45-s − 6.28·47-s + 49-s − 5.21·51-s − 0.328·53-s + 1.90·55-s + ⋯ |
| L(s) = 1 | + 0.939·3-s + 0.853·5-s − 0.377·7-s − 0.118·9-s + 0.301·11-s − 0.277·13-s + 0.801·15-s − 0.778·17-s − 0.385·19-s − 0.354·21-s − 1.31·23-s − 0.271·25-s − 1.05·27-s − 1.93·29-s + 1.97·31-s + 0.283·33-s − 0.322·35-s + 1.23·37-s − 0.260·39-s + 0.722·41-s + 1.00·43-s − 0.100·45-s − 0.917·47-s + 0.142·49-s − 0.730·51-s − 0.0451·53-s + 0.257·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 - 1.62T + 3T^{2} \) |
| 5 | \( 1 - 1.90T + 5T^{2} \) |
| 17 | \( 1 + 3.20T + 17T^{2} \) |
| 19 | \( 1 + 1.68T + 19T^{2} \) |
| 23 | \( 1 + 6.28T + 23T^{2} \) |
| 29 | \( 1 + 10.4T + 29T^{2} \) |
| 31 | \( 1 - 10.9T + 31T^{2} \) |
| 37 | \( 1 - 7.50T + 37T^{2} \) |
| 41 | \( 1 - 4.62T + 41T^{2} \) |
| 43 | \( 1 - 6.59T + 43T^{2} \) |
| 47 | \( 1 + 6.28T + 47T^{2} \) |
| 53 | \( 1 + 0.328T + 53T^{2} \) |
| 59 | \( 1 + 0.221T + 59T^{2} \) |
| 61 | \( 1 + 2.97T + 61T^{2} \) |
| 67 | \( 1 - 5.06T + 67T^{2} \) |
| 71 | \( 1 + 4.48T + 71T^{2} \) |
| 73 | \( 1 + 11.4T + 73T^{2} \) |
| 79 | \( 1 + 8.26T + 79T^{2} \) |
| 83 | \( 1 + 16.2T + 83T^{2} \) |
| 89 | \( 1 + 12.3T + 89T^{2} \) |
| 97 | \( 1 - 17.9T + 97T^{2} \) |
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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
−7.66889360904242090685001518418, −6.76439742521779500910512964545, −5.99483726740740531161503807902, −5.69680529799440054433658676649, −4.40511957590511309406049624731, −3.94782461609699127356225898855, −2.84318681768069752678120844869, −2.37637450964850415531118444975, −1.57781969362899684004733331654, 0,
1.57781969362899684004733331654, 2.37637450964850415531118444975, 2.84318681768069752678120844869, 3.94782461609699127356225898855, 4.40511957590511309406049624731, 5.69680529799440054433658676649, 5.99483726740740531161503807902, 6.76439742521779500910512964545, 7.66889360904242090685001518418
