L(s) = 1 | + 2-s + 0.987i·3-s + 4-s + (1.85 − 1.25i)5-s + 0.987i·6-s − 4.78i·7-s + 8-s + 2.02·9-s + (1.85 − 1.25i)10-s − 5.98·11-s + 0.987i·12-s + 3.49·13-s − 4.78i·14-s + (1.23 + 1.83i)15-s + 16-s − 4.96·17-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.570i·3-s + 0.5·4-s + (0.829 − 0.559i)5-s + 0.403i·6-s − 1.81i·7-s + 0.353·8-s + 0.674·9-s + (0.586 − 0.395i)10-s − 1.80·11-s + 0.285i·12-s + 0.969·13-s − 1.28i·14-s + (0.318 + 0.472i)15-s + 0.250·16-s − 1.20·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.964 + 0.264i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.964 + 0.264i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.23117 - 0.300616i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.23117 - 0.300616i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 5 | \( 1 + (-1.85 + 1.25i)T \) |
| 37 | \( 1 + (-3.96 - 4.61i)T \) |
good | 3 | \( 1 - 0.987iT - 3T^{2} \) |
| 7 | \( 1 + 4.78iT - 7T^{2} \) |
| 11 | \( 1 + 5.98T + 11T^{2} \) |
| 13 | \( 1 - 3.49T + 13T^{2} \) |
| 17 | \( 1 + 4.96T + 17T^{2} \) |
| 19 | \( 1 - 7.33iT - 19T^{2} \) |
| 23 | \( 1 - 1.74T + 23T^{2} \) |
| 29 | \( 1 - 7.85iT - 29T^{2} \) |
| 31 | \( 1 - 3.24iT - 31T^{2} \) |
| 41 | \( 1 + 0.530T + 41T^{2} \) |
| 43 | \( 1 + 1.76T + 43T^{2} \) |
| 47 | \( 1 + 4.30iT - 47T^{2} \) |
| 53 | \( 1 + 3.66iT - 53T^{2} \) |
| 59 | \( 1 + 2.15iT - 59T^{2} \) |
| 61 | \( 1 + 3.06iT - 61T^{2} \) |
| 67 | \( 1 + 3.79iT - 67T^{2} \) |
| 71 | \( 1 + 8.47T + 71T^{2} \) |
| 73 | \( 1 - 9.05iT - 73T^{2} \) |
| 79 | \( 1 + 5.56iT - 79T^{2} \) |
| 83 | \( 1 + 3.77iT - 83T^{2} \) |
| 89 | \( 1 + 8.45iT - 89T^{2} \) |
| 97 | \( 1 + 3.64T + 97T^{2} \) |
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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
−11.01912038682803575015850064491, −10.40813087938178799978309383213, −10.02155883977831589609270242173, −8.524700129239304176207466938723, −7.46530806170326629495335289006, −6.45972506187380997403812078953, −5.19645578868219650530748441278, −4.45967534570628035545044518177, −3.42642293707260447109328845837, −1.53376509954211890267685766030,
2.27042460515026142505502233922, 2.66097963354595395714270403177, 4.72357465053529621543121866349, 5.75136725446618750274972646259, 6.37408445897663903710792439895, 7.47212179864919847760130083928, 8.640023195808656264034070397819, 9.608737478237245253293202191110, 10.81623396817877609723916059085, 11.43903107105458665121583980901