L(s) = 1 | + 2-s + (0.707 − 0.707i)3-s + 4-s + (2.00 + 0.994i)5-s + (0.707 − 0.707i)6-s + (0.593 − 0.593i)7-s + 8-s − 1.00i·9-s + (2.00 + 0.994i)10-s + 2.21i·11-s + (0.707 − 0.707i)12-s − 0.163·13-s + (0.593 − 0.593i)14-s + (2.11 − 0.712i)15-s + 16-s − 5.23i·17-s + ⋯ |
L(s) = 1 | + 0.707·2-s + (0.408 − 0.408i)3-s + 0.5·4-s + (0.895 + 0.444i)5-s + (0.288 − 0.288i)6-s + (0.224 − 0.224i)7-s + 0.353·8-s − 0.333i·9-s + (0.633 + 0.314i)10-s + 0.668i·11-s + (0.204 − 0.204i)12-s − 0.0453·13-s + (0.158 − 0.158i)14-s + (0.547 − 0.184i)15-s + 0.250·16-s − 1.26i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 + 0.197i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.980 + 0.197i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.455157385\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.455157385\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-2.00 - 0.994i)T \) |
| 37 | \( 1 + (-4.07 - 4.51i)T \) |
good | 7 | \( 1 + (-0.593 + 0.593i)T - 7iT^{2} \) |
| 11 | \( 1 - 2.21iT - 11T^{2} \) |
| 13 | \( 1 + 0.163T + 13T^{2} \) |
| 17 | \( 1 + 5.23iT - 17T^{2} \) |
| 19 | \( 1 + (-1.67 - 1.67i)T + 19iT^{2} \) |
| 23 | \( 1 - 3.86T + 23T^{2} \) |
| 29 | \( 1 + (4.79 - 4.79i)T - 29iT^{2} \) |
| 31 | \( 1 + (5.64 + 5.64i)T + 31iT^{2} \) |
| 41 | \( 1 + 5.85iT - 41T^{2} \) |
| 43 | \( 1 - 8.94T + 43T^{2} \) |
| 47 | \( 1 + (7.97 - 7.97i)T - 47iT^{2} \) |
| 53 | \( 1 + (7.94 + 7.94i)T + 53iT^{2} \) |
| 59 | \( 1 + (1.00 + 1.00i)T + 59iT^{2} \) |
| 61 | \( 1 + (7.50 + 7.50i)T + 61iT^{2} \) |
| 67 | \( 1 + (5.39 + 5.39i)T + 67iT^{2} \) |
| 71 | \( 1 - 14.9T + 71T^{2} \) |
| 73 | \( 1 + (0.258 - 0.258i)T - 73iT^{2} \) |
| 79 | \( 1 + (-1.08 - 1.08i)T + 79iT^{2} \) |
| 83 | \( 1 + (4.08 + 4.08i)T + 83iT^{2} \) |
| 89 | \( 1 + (12.5 - 12.5i)T - 89iT^{2} \) |
| 97 | \( 1 - 12.7iT - 97T^{2} \) |
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
−9.515009279721412514126496045555, −9.400046737790555891650889404924, −7.82419312514091265895561466810, −7.24893209017934350533742999809, −6.49219241984688917503876951156, −5.51145622162359240961071734232, −4.71466246387816979224617453242, −3.41907918580721683833394424842, −2.52614656677087951585101940835, −1.51030524753154121247912174669,
1.53227430663416436600888738171, 2.64963490793503598238019441012, 3.69822490138618060861348673119, 4.71472454612585943063568461134, 5.56943372982286295872331822992, 6.16151576580222310106213801667, 7.34479150474517082167939307671, 8.378590241739177033845823964586, 9.045641679441632742048162489245, 9.835108645387569235835292280099