L(s) = 1 | + (−0.707 − 0.707i)2-s + 1.00i·4-s + (−0.707 + 0.292i)5-s + (0.707 − 0.707i)8-s + (−0.707 + 0.707i)9-s + (0.707 + 0.292i)10-s − 1.00·16-s + (−0.707 − 0.707i)17-s + 1.00·18-s + (−0.292 − 0.707i)20-s + (−0.292 + 0.292i)25-s + (0.707 − 0.292i)29-s + (0.707 + 0.707i)32-s + 1.00i·34-s + (−0.707 − 0.707i)36-s + (0.292 + 0.707i)37-s + ⋯ |
L(s) = 1 | + (−0.707 − 0.707i)2-s + 1.00i·4-s + (−0.707 + 0.292i)5-s + (0.707 − 0.707i)8-s + (−0.707 + 0.707i)9-s + (0.707 + 0.292i)10-s − 1.00·16-s + (−0.707 − 0.707i)17-s + 1.00·18-s + (−0.292 − 0.707i)20-s + (−0.292 + 0.292i)25-s + (0.707 − 0.292i)29-s + (0.707 + 0.707i)32-s + 1.00i·34-s + (−0.707 − 0.707i)36-s + (0.292 + 0.707i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0758i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0758i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.07389166305\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.07389166305\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.707 + 0.707i)T \) |
| 7 | \( 1 \) |
| 17 | \( 1 + (0.707 + 0.707i)T \) |
good | 3 | \( 1 + (0.707 - 0.707i)T^{2} \) |
| 5 | \( 1 + (0.707 - 0.292i)T + (0.707 - 0.707i)T^{2} \) |
| 11 | \( 1 + (0.707 + 0.707i)T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 19 | \( 1 - iT^{2} \) |
| 23 | \( 1 + (0.707 + 0.707i)T^{2} \) |
| 29 | \( 1 + (-0.707 + 0.292i)T + (0.707 - 0.707i)T^{2} \) |
| 31 | \( 1 + (0.707 - 0.707i)T^{2} \) |
| 37 | \( 1 + (-0.292 - 0.707i)T + (-0.707 + 0.707i)T^{2} \) |
| 41 | \( 1 + (1.70 + 0.707i)T + (0.707 + 0.707i)T^{2} \) |
| 43 | \( 1 + iT^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( 1 + (1 + i)T + iT^{2} \) |
| 59 | \( 1 + iT^{2} \) |
| 61 | \( 1 + (1.70 + 0.707i)T + (0.707 + 0.707i)T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 + (0.707 - 0.707i)T^{2} \) |
| 73 | \( 1 + (-0.707 + 0.292i)T + (0.707 - 0.707i)T^{2} \) |
| 79 | \( 1 + (0.707 + 0.707i)T^{2} \) |
| 83 | \( 1 - iT^{2} \) |
| 89 | \( 1 + 2iT - T^{2} \) |
| 97 | \( 1 + (0.707 - 0.292i)T + (0.707 - 0.707i)T^{2} \) |
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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
−8.350041794604747534556528360058, −7.958827609664927979774669611785, −7.14897347051146865434649994703, −6.44588762522140084105444935730, −5.16696007834147954951335190888, −4.42526781571625400715926551456, −3.41153363763487267252783706771, −2.75360915753655870782156684350, −1.75546052181130333540948532299, −0.05679734376447411383845962127,
1.37071762089987271138999012204, 2.71118388306182570542154652944, 3.89605288478848855727840657011, 4.67807656195524327384487625290, 5.61288459825055659886386959687, 6.37254457185783227428583283178, 6.91064784170646099688086147114, 7.994433313080060375173902418839, 8.281447061158142341916447947615, 9.054319095608805320200579810290