L(s) = 1 | + (0.923 − 0.382i)5-s + i·9-s + (0.541 + 0.541i)13-s + (−1.30 + 1.30i)17-s + (0.707 − 0.707i)25-s + 1.41i·29-s + (1.41 + 1.41i)37-s − 0.765i·41-s + (0.382 + 0.923i)45-s + (−1 + i)53-s − 1.84i·61-s + (0.707 + 0.292i)65-s + (−1.30 − 1.30i)73-s − 81-s + (−0.707 + 1.70i)85-s + ⋯ |
L(s) = 1 | + (0.923 − 0.382i)5-s + i·9-s + (0.541 + 0.541i)13-s + (−1.30 + 1.30i)17-s + (0.707 − 0.707i)25-s + 1.41i·29-s + (1.41 + 1.41i)37-s − 0.765i·41-s + (0.382 + 0.923i)45-s + (−1 + i)53-s − 1.84i·61-s + (0.707 + 0.292i)65-s + (−1.30 − 1.30i)73-s − 81-s + (−0.707 + 1.70i)85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.704 - 0.709i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.704 - 0.709i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.483887867\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.483887867\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-0.923 + 0.382i)T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 - iT^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 + (-0.541 - 0.541i)T + iT^{2} \) |
| 17 | \( 1 + (1.30 - 1.30i)T - iT^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + iT^{2} \) |
| 29 | \( 1 - 1.41iT - T^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 + (-1.41 - 1.41i)T + iT^{2} \) |
| 41 | \( 1 + 0.765iT - T^{2} \) |
| 43 | \( 1 + iT^{2} \) |
| 47 | \( 1 + iT^{2} \) |
| 53 | \( 1 + (1 - i)T - iT^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 + 1.84iT - T^{2} \) |
| 67 | \( 1 - iT^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + (1.30 + 1.30i)T + iT^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 - iT^{2} \) |
| 89 | \( 1 - 1.84T + T^{2} \) |
| 97 | \( 1 + (-0.541 + 0.541i)T - iT^{2} \) |
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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
−8.688611884882502080297065120248, −8.229478655216381607833906830019, −7.22467916138317068368960589988, −6.34361830506001743914063444148, −5.93230159192083491690770657785, −4.84218862036974696461527659015, −4.45486307954268323632521299033, −3.21721838160645157858781019448, −2.09412556922186869735400329816, −1.52966162116759110982921459493,
0.860143963808678002473206125729, 2.24712535752377705102801524397, 2.91223489637797268062145052274, 3.93458766718382285327602991611, 4.80117939744244548801327841661, 5.82389687482335868153518482127, 6.25346788586994715659972515086, 6.97146316509373695642585257973, 7.74238613772110780451369992948, 8.773838263378353436824161751871