| L(s) = 1 | + (1 + i)5-s + (1 − i)13-s + 2·17-s − 3i·25-s + (3 − 3i)29-s + (−5 − 5i)37-s − 10i·41-s + 7·49-s + (9 + 9i)53-s + (−1 + i)61-s + 2·65-s + 6i·73-s + (2 + 2i)85-s − 16i·89-s − 8·97-s + ⋯ |
| L(s) = 1 | + (0.447 + 0.447i)5-s + (0.277 − 0.277i)13-s + 0.485·17-s − 0.600i·25-s + (0.557 − 0.557i)29-s + (−0.821 − 0.821i)37-s − 1.56i·41-s + 49-s + (1.23 + 1.23i)53-s + (−0.128 + 0.128i)61-s + 0.248·65-s + 0.702i·73-s + (0.216 + 0.216i)85-s − 1.69i·89-s − 0.812·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 + 0.382i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.923 + 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.109611700\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.109611700\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (-1 - i)T + 5iT^{2} \) |
| 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 + 11iT^{2} \) |
| 13 | \( 1 + (-1 + i)T - 13iT^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 19 | \( 1 - 19iT^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 + (-3 + 3i)T - 29iT^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + (5 + 5i)T + 37iT^{2} \) |
| 41 | \( 1 + 10iT - 41T^{2} \) |
| 43 | \( 1 + 43iT^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + (-9 - 9i)T + 53iT^{2} \) |
| 59 | \( 1 + 59iT^{2} \) |
| 61 | \( 1 + (1 - i)T - 61iT^{2} \) |
| 67 | \( 1 - 67iT^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 - 6iT - 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 83iT^{2} \) |
| 89 | \( 1 + 16iT - 89T^{2} \) |
| 97 | \( 1 + 8T + 97T^{2} \) |
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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
−8.350729149007756589859324713126, −7.42131480778202259531747235694, −6.91438854938860765028197656955, −5.92745409209069382125638039820, −5.59013336230329036198595987274, −4.48973493160142592018787505122, −3.69588338593725878564570192262, −2.77162956573359561061413093783, −1.98154852064004869687644201864, −0.69348739381659912831829692192,
0.995432605139636778114986141510, 1.86768030491735816623022548208, 2.98295216442246247129436463320, 3.79806402014332806114181623906, 4.79940927341321286917146669003, 5.35194630713359224657037072365, 6.18132580012906941399980901513, 6.86882153217375179451994709921, 7.66481479287116849880675349808, 8.495990798371836682178950568105
