| L(s) = 1 | − i·2-s − i·3-s − 4-s − 6-s + 0.511i·7-s + i·8-s − 9-s + 3.29·11-s + i·12-s − 2.65i·13-s + 0.511·14-s + 16-s + 5.57i·17-s + i·18-s − 0.374·19-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.577i·3-s − 0.5·4-s − 0.408·6-s + 0.193i·7-s + 0.353i·8-s − 0.333·9-s + 0.993·11-s + 0.288i·12-s − 0.736i·13-s + 0.136·14-s + 0.250·16-s + 1.35i·17-s + 0.235i·18-s − 0.0858·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3750 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3750 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.820905417\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.820905417\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + iT \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 0.511iT - 7T^{2} \) |
| 11 | \( 1 - 3.29T + 11T^{2} \) |
| 13 | \( 1 + 2.65iT - 13T^{2} \) |
| 17 | \( 1 - 5.57iT - 17T^{2} \) |
| 19 | \( 1 + 0.374T + 19T^{2} \) |
| 23 | \( 1 + 5.75iT - 23T^{2} \) |
| 29 | \( 1 - 3.89T + 29T^{2} \) |
| 31 | \( 1 + 1.80T + 31T^{2} \) |
| 37 | \( 1 - 10.3iT - 37T^{2} \) |
| 41 | \( 1 - 2.33T + 41T^{2} \) |
| 43 | \( 1 - 5.87iT - 43T^{2} \) |
| 47 | \( 1 + 8.19iT - 47T^{2} \) |
| 53 | \( 1 + 0.518iT - 53T^{2} \) |
| 59 | \( 1 - 7.62T + 59T^{2} \) |
| 61 | \( 1 - 2.90T + 61T^{2} \) |
| 67 | \( 1 + 1.78iT - 67T^{2} \) |
| 71 | \( 1 - 7.27T + 71T^{2} \) |
| 73 | \( 1 - 1.40iT - 73T^{2} \) |
| 79 | \( 1 - 12.1T + 79T^{2} \) |
| 83 | \( 1 + 3.09iT - 83T^{2} \) |
| 89 | \( 1 + 3.42T + 89T^{2} \) |
| 97 | \( 1 + 15.1iT - 97T^{2} \) |
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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
−8.495379901431532327457101873182, −7.78925011155464039722389356238, −6.67276554530642036611577521353, −6.23186368191093430159511716465, −5.29288665391279195384666230669, −4.34835077029204489018799552978, −3.55157965890240636772148327746, −2.64335299876415013157161082421, −1.70821901860594511549881580449, −0.75415737860635199613155401933,
0.870689289864353836958925763904, 2.30881180570724784172698498669, 3.58655444174073437338873957647, 4.12086573791115325664611324950, 4.99074108151958188207324755906, 5.66975451478769541528546961470, 6.56212099825053014595671521474, 7.17958029264428344252552077613, 7.82752292415562850082480615547, 8.996133401146336027493674689728
