L(s) = 1 | − i·3-s − 1.57i·7-s − 9-s + 3.88·11-s − 0.343i·13-s + 6.07i·17-s − 3.69·19-s − 1.57·21-s + 1.39i·23-s + i·27-s + 3.32·29-s − 5.25·31-s − 3.88i·33-s + 8.56i·37-s − 0.343·39-s + ⋯ |
L(s) = 1 | − 0.577i·3-s − 0.596i·7-s − 0.333·9-s + 1.17·11-s − 0.0952i·13-s + 1.47i·17-s − 0.846·19-s − 0.344·21-s + 0.289i·23-s + 0.192i·27-s + 0.618·29-s − 0.943·31-s − 0.675i·33-s + 1.40i·37-s − 0.0549·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7500 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9490912436\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9490912436\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + 1.57iT - 7T^{2} \) |
| 11 | \( 1 - 3.88T + 11T^{2} \) |
| 13 | \( 1 + 0.343iT - 13T^{2} \) |
| 17 | \( 1 - 6.07iT - 17T^{2} \) |
| 19 | \( 1 + 3.69T + 19T^{2} \) |
| 23 | \( 1 - 1.39iT - 23T^{2} \) |
| 29 | \( 1 - 3.32T + 29T^{2} \) |
| 31 | \( 1 + 5.25T + 31T^{2} \) |
| 37 | \( 1 - 8.56iT - 37T^{2} \) |
| 41 | \( 1 + 1.27T + 41T^{2} \) |
| 43 | \( 1 - 1.42iT - 43T^{2} \) |
| 47 | \( 1 - 0.375iT - 47T^{2} \) |
| 53 | \( 1 + 11.2iT - 53T^{2} \) |
| 59 | \( 1 + 11.5T + 59T^{2} \) |
| 61 | \( 1 + 10.9T + 61T^{2} \) |
| 67 | \( 1 - 10.4iT - 67T^{2} \) |
| 71 | \( 1 + 10.1T + 71T^{2} \) |
| 73 | \( 1 - 13.1iT - 73T^{2} \) |
| 79 | \( 1 + 13.7T + 79T^{2} \) |
| 83 | \( 1 + 4.62iT - 83T^{2} \) |
| 89 | \( 1 - 7.26T + 89T^{2} \) |
| 97 | \( 1 - 6.05iT - 97T^{2} \) |
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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
−8.037168810434472199984596118059, −7.34711180189850790542903778719, −6.51406398132922047311916665388, −6.30583275002899544952393836445, −5.35295140565628591179470760982, −4.31014364749422526455125496384, −3.84850967591157169343202464495, −2.93106327623728960019079629690, −1.74839725355197382556541851432, −1.21666464501048701883911814256,
0.22511926489079059664572146396, 1.60357872993855803775456593766, 2.58545945421306852532877592760, 3.32993626978917596200677254719, 4.27896186925909090315350119377, 4.71679228771167038428529404664, 5.69331370643548957655463437109, 6.19305675497956137084120242700, 7.03785272082885420531773067656, 7.65161318240028123937892402387