L(s) = 1 | − 0.765i·2-s + 0.414·4-s + (−0.707 + 0.707i)5-s + (−0.541 − 0.541i)7-s − 1.08i·8-s − i·9-s + (0.541 + 0.541i)10-s + (0.707 + 0.707i)11-s + 1.84·13-s + (−0.414 + 0.414i)14-s − 0.414·16-s + (−0.923 − 0.382i)17-s − 0.765·18-s + (−0.292 + 0.292i)20-s + (0.541 − 0.541i)22-s + ⋯ |
L(s) = 1 | − 0.765i·2-s + 0.414·4-s + (−0.707 + 0.707i)5-s + (−0.541 − 0.541i)7-s − 1.08i·8-s − i·9-s + (0.541 + 0.541i)10-s + (0.707 + 0.707i)11-s + 1.84·13-s + (−0.414 + 0.414i)14-s − 0.414·16-s + (−0.923 − 0.382i)17-s − 0.765·18-s + (−0.292 + 0.292i)20-s + (0.541 − 0.541i)22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 935 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.266 + 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 935 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.266 + 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.060350295\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.060350295\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (0.707 - 0.707i)T \) |
| 11 | \( 1 + (-0.707 - 0.707i)T \) |
| 17 | \( 1 + (0.923 + 0.382i)T \) |
good | 2 | \( 1 + 0.765iT - T^{2} \) |
| 3 | \( 1 + iT^{2} \) |
| 7 | \( 1 + (0.541 + 0.541i)T + iT^{2} \) |
| 13 | \( 1 - 1.84T + T^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 - iT^{2} \) |
| 29 | \( 1 + iT^{2} \) |
| 31 | \( 1 - iT^{2} \) |
| 37 | \( 1 + iT^{2} \) |
| 41 | \( 1 - iT^{2} \) |
| 43 | \( 1 - 0.765iT - T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 - 2iT - T^{2} \) |
| 61 | \( 1 - iT^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 + (-1 + i)T - iT^{2} \) |
| 73 | \( 1 + (1.30 - 1.30i)T - iT^{2} \) |
| 79 | \( 1 - iT^{2} \) |
| 83 | \( 1 - 1.84iT - T^{2} \) |
| 89 | \( 1 + 1.41T + T^{2} \) |
| 97 | \( 1 + iT^{2} \) |
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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
−10.26858837473222673843427098829, −9.472360987686806592673338840706, −8.590562817325825252622282581161, −7.29423811285609830876692841065, −6.66497512866833516332149050026, −6.20691999669284669650620818760, −4.08679231780695389037261176480, −3.76374699999811855591285506526, −2.76773362579967433874241577313, −1.19857306423339178745336923668,
1.71979303738022877769803329073, 3.22960695834670065310139284529, 4.30768374444600407889221178591, 5.49941140168612839077781225583, 6.15323852754457268973539689359, 6.95787255923672607913784716699, 8.123557834212673617754308756073, 8.495490608231397907191089536091, 9.182450805629920632643843133280, 10.74328427695596704679770491998