L(s) = 1 | + (0.923 + 0.382i)2-s + (−1.30 − 1.30i)3-s + (0.707 + 0.707i)4-s + (−0.707 − 1.70i)6-s + (0.382 + 0.923i)8-s + 2.41i·9-s + 1.41i·11-s − 1.84i·12-s + (−1.30 + 1.30i)13-s + i·16-s + (−0.923 + 2.23i)18-s − 19-s + (−0.541 + 1.30i)22-s + (0.707 − 1.70i)24-s + (−1.70 + 0.707i)26-s + (1.84 − 1.84i)27-s + ⋯ |
L(s) = 1 | + (0.923 + 0.382i)2-s + (−1.30 − 1.30i)3-s + (0.707 + 0.707i)4-s + (−0.707 − 1.70i)6-s + (0.382 + 0.923i)8-s + 2.41i·9-s + 1.41i·11-s − 1.84i·12-s + (−1.30 + 1.30i)13-s + i·16-s + (−0.923 + 2.23i)18-s − 19-s + (−0.541 + 1.30i)22-s + (0.707 − 1.70i)24-s + (−1.70 + 0.707i)26-s + (1.84 − 1.84i)27-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.229 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.229 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.010232332\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.010232332\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.923 - 0.382i)T \) |
| 5 | \( 1 \) |
| 19 | \( 1 + T \) |
good | 3 | \( 1 + (1.30 + 1.30i)T + iT^{2} \) |
| 7 | \( 1 + iT^{2} \) |
| 11 | \( 1 - 1.41iT - T^{2} \) |
| 13 | \( 1 + (1.30 - 1.30i)T - iT^{2} \) |
| 17 | \( 1 - iT^{2} \) |
| 23 | \( 1 - iT^{2} \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 + (-0.541 - 0.541i)T + iT^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - iT^{2} \) |
| 47 | \( 1 + iT^{2} \) |
| 53 | \( 1 + (-0.541 + 0.541i)T - iT^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - 1.41T + T^{2} \) |
| 67 | \( 1 + (0.541 - 0.541i)T - iT^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + iT^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - iT^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 + (-0.541 - 0.541i)T + iT^{2} \) |
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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
−9.725284115061841409299680301369, −8.375152746993087340521944653875, −7.48669257367086405767333819839, −6.89937560559978576226876197866, −6.62812838172203474209261624383, −5.59987524593255436979246764292, −4.81708959969662501035446230896, −4.28196597036317042190303012236, −2.35463293718713129196067622440, −1.85218336235448255564326332719,
0.61325165660979778554563150180, 2.71278323377652451438799651603, 3.58487533245122570704102365409, 4.42998549005147472519547947383, 5.16557802297764306155592322587, 5.78943032770958734435108918758, 6.29208410174247330503195967262, 7.42205601884610892574209799138, 8.675227021650885560463504803050, 9.722267322658238522240236291688