| L(s) = 1 | + 3.32·3-s + 2.67·5-s + 7-s + 8.08·9-s + 11-s − 13-s + 8.92·15-s − 3.70·17-s − 5.63·19-s + 3.32·21-s + 3.88·23-s + 2.18·25-s + 16.9·27-s + 7.80·29-s − 9.28·31-s + 3.32·33-s + 2.67·35-s − 5.64·37-s − 3.32·39-s − 2.71·41-s + 7.67·43-s + 21.6·45-s + 7.32·47-s + 49-s − 12.3·51-s + 5.24·53-s + 2.67·55-s + ⋯ |
| L(s) = 1 | + 1.92·3-s + 1.19·5-s + 0.377·7-s + 2.69·9-s + 0.301·11-s − 0.277·13-s + 2.30·15-s − 0.897·17-s − 1.29·19-s + 0.726·21-s + 0.809·23-s + 0.436·25-s + 3.25·27-s + 1.44·29-s − 1.66·31-s + 0.579·33-s + 0.452·35-s − 0.927·37-s − 0.533·39-s − 0.423·41-s + 1.17·43-s + 3.22·45-s + 1.06·47-s + 0.142·49-s − 1.72·51-s + 0.720·53-s + 0.361·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.297854036\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.297854036\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 3.32T + 3T^{2} \) |
| 5 | \( 1 - 2.67T + 5T^{2} \) |
| 17 | \( 1 + 3.70T + 17T^{2} \) |
| 19 | \( 1 + 5.63T + 19T^{2} \) |
| 23 | \( 1 - 3.88T + 23T^{2} \) |
| 29 | \( 1 - 7.80T + 29T^{2} \) |
| 31 | \( 1 + 9.28T + 31T^{2} \) |
| 37 | \( 1 + 5.64T + 37T^{2} \) |
| 41 | \( 1 + 2.71T + 41T^{2} \) |
| 43 | \( 1 - 7.67T + 43T^{2} \) |
| 47 | \( 1 - 7.32T + 47T^{2} \) |
| 53 | \( 1 - 5.24T + 53T^{2} \) |
| 59 | \( 1 - 4.28T + 59T^{2} \) |
| 61 | \( 1 + 1.00T + 61T^{2} \) |
| 67 | \( 1 + 13.1T + 67T^{2} \) |
| 71 | \( 1 - 10.8T + 71T^{2} \) |
| 73 | \( 1 - 6.70T + 73T^{2} \) |
| 79 | \( 1 + 8.98T + 79T^{2} \) |
| 83 | \( 1 - 5.93T + 83T^{2} \) |
| 89 | \( 1 - 5.66T + 89T^{2} \) |
| 97 | \( 1 + 2.80T + 97T^{2} \) |
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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
−8.740945240903888526639911798538, −7.87800773939880780483357162874, −7.02285872654727325398144266256, −6.53864614704018950863657923876, −5.37048350896925627957066140178, −4.44773316883996879193271984086, −3.77988669882698621862548328584, −2.63701849096992149493706604659, −2.20502514087907679249739438768, −1.40584260163237271505371678574,
1.40584260163237271505371678574, 2.20502514087907679249739438768, 2.63701849096992149493706604659, 3.77988669882698621862548328584, 4.44773316883996879193271984086, 5.37048350896925627957066140178, 6.53864614704018950863657923876, 7.02285872654727325398144266256, 7.87800773939880780483357162874, 8.740945240903888526639911798538
