L(s) = 1 | − 2-s + (0.707 + 0.707i)3-s + 4-s + (−2.02 − 0.943i)5-s + (−0.707 − 0.707i)6-s + (−0.516 − 0.516i)7-s − 8-s + 1.00i·9-s + (2.02 + 0.943i)10-s + 0.497i·11-s + (0.707 + 0.707i)12-s − 1.53·13-s + (0.516 + 0.516i)14-s + (−0.766 − 2.10i)15-s + 16-s + 2.55i·17-s + ⋯ |
L(s) = 1 | − 0.707·2-s + (0.408 + 0.408i)3-s + 0.5·4-s + (−0.906 − 0.421i)5-s + (−0.288 − 0.288i)6-s + (−0.195 − 0.195i)7-s − 0.353·8-s + 0.333i·9-s + (0.641 + 0.298i)10-s + 0.149i·11-s + (0.204 + 0.204i)12-s − 0.424·13-s + (0.137 + 0.137i)14-s + (−0.197 − 0.542i)15-s + 0.250·16-s + 0.619i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.673 + 0.739i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.673 + 0.739i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9309930498\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9309930498\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 + (-0.707 - 0.707i)T \) |
| 5 | \( 1 + (2.02 + 0.943i)T \) |
| 37 | \( 1 + (-4.64 + 3.93i)T \) |
good | 7 | \( 1 + (0.516 + 0.516i)T + 7iT^{2} \) |
| 11 | \( 1 - 0.497iT - 11T^{2} \) |
| 13 | \( 1 + 1.53T + 13T^{2} \) |
| 17 | \( 1 - 2.55iT - 17T^{2} \) |
| 19 | \( 1 + (-2.36 + 2.36i)T - 19iT^{2} \) |
| 23 | \( 1 - 5.13T + 23T^{2} \) |
| 29 | \( 1 + (7.56 + 7.56i)T + 29iT^{2} \) |
| 31 | \( 1 + (-5.92 + 5.92i)T - 31iT^{2} \) |
| 41 | \( 1 - 2.70iT - 41T^{2} \) |
| 43 | \( 1 - 9.05T + 43T^{2} \) |
| 47 | \( 1 + (9.26 + 9.26i)T + 47iT^{2} \) |
| 53 | \( 1 + (0.413 - 0.413i)T - 53iT^{2} \) |
| 59 | \( 1 + (-5.91 + 5.91i)T - 59iT^{2} \) |
| 61 | \( 1 + (-10.7 + 10.7i)T - 61iT^{2} \) |
| 67 | \( 1 + (9.29 - 9.29i)T - 67iT^{2} \) |
| 71 | \( 1 + 4.10T + 71T^{2} \) |
| 73 | \( 1 + (2.63 + 2.63i)T + 73iT^{2} \) |
| 79 | \( 1 + (0.827 - 0.827i)T - 79iT^{2} \) |
| 83 | \( 1 + (-11.2 + 11.2i)T - 83iT^{2} \) |
| 89 | \( 1 + (-6.24 - 6.24i)T + 89iT^{2} \) |
| 97 | \( 1 - 3.13iT - 97T^{2} \) |
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
−9.569689180540617294233647752665, −9.020680237446006550752720687563, −8.031845217046993916120501545079, −7.60546686565475214159441527142, −6.64164902839562000084005373443, −5.38138005674280262683165644484, −4.34429121447940162389163787307, −3.48872665675930947327918422527, −2.28664987087633750713533077299, −0.58577271535074833162145820315,
1.12233743422713703299977274235, 2.72368081860261818147910316792, 3.34763853898715103724798971643, 4.71588185876787371066370106118, 5.96631397697178728713020018795, 7.09193634996875701363658910395, 7.38234287805761258737240990393, 8.312359040168751117151585211522, 9.039265610485041319837603387943, 9.797218479061382115580196191071