L(s) = 1 | + (−0.623 + 0.781i)3-s + (0.222 − 0.974i)7-s + (−0.222 − 0.974i)9-s + (−0.277 + 1.21i)13-s + 0.445·19-s + (0.623 + 0.781i)21-s + (−0.222 − 0.974i)25-s + (0.900 + 0.433i)27-s + 1.80·31-s + (1.62 − 0.781i)37-s + (−0.777 − 0.974i)39-s + (0.277 + 0.347i)43-s + (−0.900 − 0.433i)49-s + (−0.277 + 0.347i)57-s + (1.62 − 0.781i)61-s + ⋯ |
L(s) = 1 | + (−0.623 + 0.781i)3-s + (0.222 − 0.974i)7-s + (−0.222 − 0.974i)9-s + (−0.277 + 1.21i)13-s + 0.445·19-s + (0.623 + 0.781i)21-s + (−0.222 − 0.974i)25-s + (0.900 + 0.433i)27-s + 1.80·31-s + (1.62 − 0.781i)37-s + (−0.777 − 0.974i)39-s + (0.277 + 0.347i)43-s + (−0.900 − 0.433i)49-s + (−0.277 + 0.347i)57-s + (1.62 − 0.781i)61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.967 - 0.253i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.967 - 0.253i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9810123462\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9810123462\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.623 - 0.781i)T \) |
| 7 | \( 1 + (-0.222 + 0.974i)T \) |
good | 5 | \( 1 + (0.222 + 0.974i)T^{2} \) |
| 11 | \( 1 + (0.900 + 0.433i)T^{2} \) |
| 13 | \( 1 + (0.277 - 1.21i)T + (-0.900 - 0.433i)T^{2} \) |
| 17 | \( 1 + (-0.623 + 0.781i)T^{2} \) |
| 19 | \( 1 - 0.445T + T^{2} \) |
| 23 | \( 1 + (-0.623 - 0.781i)T^{2} \) |
| 29 | \( 1 + (-0.623 + 0.781i)T^{2} \) |
| 31 | \( 1 - 1.80T + T^{2} \) |
| 37 | \( 1 + (-1.62 + 0.781i)T + (0.623 - 0.781i)T^{2} \) |
| 41 | \( 1 + (0.222 + 0.974i)T^{2} \) |
| 43 | \( 1 + (-0.277 - 0.347i)T + (-0.222 + 0.974i)T^{2} \) |
| 47 | \( 1 + (0.900 + 0.433i)T^{2} \) |
| 53 | \( 1 + (-0.623 - 0.781i)T^{2} \) |
| 59 | \( 1 + (0.222 - 0.974i)T^{2} \) |
| 61 | \( 1 + (-1.62 + 0.781i)T + (0.623 - 0.781i)T^{2} \) |
| 67 | \( 1 + 1.24T + T^{2} \) |
| 71 | \( 1 + (-0.623 - 0.781i)T^{2} \) |
| 73 | \( 1 + (-0.400 - 1.75i)T + (-0.900 + 0.433i)T^{2} \) |
| 79 | \( 1 - 1.80T + T^{2} \) |
| 83 | \( 1 + (0.900 - 0.433i)T^{2} \) |
| 89 | \( 1 + (0.900 - 0.433i)T^{2} \) |
| 97 | \( 1 + 0.445T + T^{2} \) |
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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
−9.495623118733570686408290785513, −8.481860703345055961768587681375, −7.63952126258834031076049359560, −6.70649066623763126917224191507, −6.19055809328985292660186625024, −5.05255331506990352096867350174, −4.37111559939004713971391963156, −3.84229710000926466984221383408, −2.53617662027461825512937351681, −0.960962460042154568664335431842,
1.06123945941217682619177959914, 2.34216653912168372872321608434, 3.10143139114688824803716569439, 4.63435876105324579654945342183, 5.37796423260338461285520213676, 5.95307713932947613769477302519, 6.73191977137187420163090464040, 7.80520001814172966188010211259, 8.055701753817002127786677435889, 9.070912536179837772652979918384