| L(s) = 1 | − i·2-s − i·3-s − 4-s − 6-s − 2i·7-s + i·8-s − 9-s − 4·11-s + i·12-s + 4i·13-s − 2·14-s + 16-s + 2i·17-s + i·18-s + 8·19-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.577i·3-s − 0.5·4-s − 0.408·6-s − 0.755i·7-s + 0.353i·8-s − 0.333·9-s − 1.20·11-s + 0.288i·12-s + 1.10i·13-s − 0.534·14-s + 0.250·16-s + 0.485i·17-s + 0.235i·18-s + 1.83·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.481151827\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.481151827\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + iT \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 \) |
| 31 | \( 1 + T \) |
| good | 7 | \( 1 + 2iT - 7T^{2} \) |
| 11 | \( 1 + 4T + 11T^{2} \) |
| 13 | \( 1 - 4iT - 13T^{2} \) |
| 17 | \( 1 - 2iT - 17T^{2} \) |
| 19 | \( 1 - 8T + 19T^{2} \) |
| 23 | \( 1 - 8iT - 23T^{2} \) |
| 29 | \( 1 + 4T + 29T^{2} \) |
| 37 | \( 1 + 12iT - 37T^{2} \) |
| 41 | \( 1 - 10T + 41T^{2} \) |
| 43 | \( 1 + 8iT - 43T^{2} \) |
| 47 | \( 1 + 4iT - 47T^{2} \) |
| 53 | \( 1 + 6iT - 53T^{2} \) |
| 59 | \( 1 + 2T + 59T^{2} \) |
| 61 | \( 1 - 10T + 61T^{2} \) |
| 67 | \( 1 + 6iT - 67T^{2} \) |
| 71 | \( 1 - 6T + 71T^{2} \) |
| 73 | \( 1 - 4iT - 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 + 4iT - 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + 18iT - 97T^{2} \) |
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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
−7.79474499584685372124096429642, −7.55469324827774005765360650381, −6.85402286405923063101260505905, −5.53517819888084463229888937213, −5.34774663777608683309726148131, −4.01323396372599657878475309241, −3.54205588717333181559432991956, −2.41655537565091887090984948153, −1.64611016318658122515897498814, −0.56412572311500521373854371210,
0.820087439455800249058163549542, 2.69960666911064440478899451637, 2.99764760815535189457164917751, 4.27825208803218609487539824259, 5.16409982133788050625785887309, 5.43450374802439849946089885545, 6.20685935731196599620090046138, 7.19601327065189032321534257337, 8.015354101477400382637630881840, 8.253107559966398113618215985478
