| L(s) = 1 | − i·2-s + 0.633·3-s − 4-s − 0.633i·6-s + (0.961 + 0.961i)7-s + i·8-s − 2.59·9-s + (−1.43 − 1.43i)11-s − 0.633·12-s + (3.31 + 3.31i)13-s + (0.961 − 0.961i)14-s + 16-s − 1.20i·17-s + 2.59i·18-s + (4.29 − 4.29i)19-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s + 0.365·3-s − 0.5·4-s − 0.258i·6-s + (0.363 + 0.363i)7-s + 0.353i·8-s − 0.866·9-s + (−0.431 − 0.431i)11-s − 0.182·12-s + (0.920 + 0.920i)13-s + (0.256 − 0.256i)14-s + 0.250·16-s − 0.292i·17-s + 0.612i·18-s + (0.986 − 0.986i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.594 + 0.804i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.594 + 0.804i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.812859820\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.812859820\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + iT \) |
| 5 | \( 1 \) |
| 29 | \( 1 + (-2.90 + 4.53i)T \) |
| good | 3 | \( 1 - 0.633T + 3T^{2} \) |
| 7 | \( 1 + (-0.961 - 0.961i)T + 7iT^{2} \) |
| 11 | \( 1 + (1.43 + 1.43i)T + 11iT^{2} \) |
| 13 | \( 1 + (-3.31 - 3.31i)T + 13iT^{2} \) |
| 17 | \( 1 + 1.20iT - 17T^{2} \) |
| 19 | \( 1 + (-4.29 + 4.29i)T - 19iT^{2} \) |
| 23 | \( 1 + (-0.821 + 0.821i)T - 23iT^{2} \) |
| 31 | \( 1 + (-6.99 - 6.99i)T + 31iT^{2} \) |
| 37 | \( 1 - 7.62T + 37T^{2} \) |
| 41 | \( 1 + (-1.02 + 1.02i)T - 41iT^{2} \) |
| 43 | \( 1 - 1.56T + 43T^{2} \) |
| 47 | \( 1 - 5.15T + 47T^{2} \) |
| 53 | \( 1 + (-6.99 + 6.99i)T - 53iT^{2} \) |
| 59 | \( 1 + 0.695iT - 59T^{2} \) |
| 61 | \( 1 + (-0.821 - 0.821i)T + 61iT^{2} \) |
| 67 | \( 1 + (9.48 - 9.48i)T - 67iT^{2} \) |
| 71 | \( 1 - 6.49iT - 71T^{2} \) |
| 73 | \( 1 - 7.91iT - 73T^{2} \) |
| 79 | \( 1 + (-8.50 + 8.50i)T - 79iT^{2} \) |
| 83 | \( 1 + (-1 + i)T - 83iT^{2} \) |
| 89 | \( 1 + (9.73 - 9.73i)T - 89iT^{2} \) |
| 97 | \( 1 + 2.86T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.313162947326486798407278850556, −8.641498469684408373152398904099, −8.206478988903187000982420776857, −7.01891214487603693941589932042, −5.96057947140903716343737399104, −5.13989035572464864365079241319, −4.17101317344356920953743185274, −3.03584414989405475988866046404, −2.41690665338965338281321164007, −0.936684669241876736118130492489,
1.03999093383918903166549917810, 2.71202288011374665826810132334, 3.66394881429150516561112340260, 4.69988100609565775436924567089, 5.71427459413128169963905484296, 6.18519581179558348585907977000, 7.60298078398744623361808442181, 7.85120503153709623995277963630, 8.635226654878628554097348101781, 9.482484738221290633598566631623
