L(s) = 1 | − 2-s + 4-s − 2.43·7-s − 8-s − 4.08·11-s + 3.79·13-s + 2.43·14-s + 16-s + 3.73·17-s + 19-s + 4.08·22-s + 0.351·23-s − 3.79·26-s − 2.43·28-s − 6.73·29-s + 9.34·31-s − 32-s − 3.73·34-s − 6.17·37-s − 38-s − 4.05·41-s + 0.913·43-s − 4.08·44-s − 0.351·46-s − 5.52·47-s − 1.05·49-s + 3.79·52-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.5·4-s − 0.921·7-s − 0.353·8-s − 1.23·11-s + 1.05·13-s + 0.651·14-s + 0.250·16-s + 0.905·17-s + 0.229·19-s + 0.871·22-s + 0.0733·23-s − 0.743·26-s − 0.460·28-s − 1.25·29-s + 1.67·31-s − 0.176·32-s − 0.640·34-s − 1.01·37-s − 0.162·38-s − 0.633·41-s + 0.139·43-s − 0.616·44-s − 0.0518·46-s − 0.805·47-s − 0.150·49-s + 0.525·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
good | 7 | \( 1 + 2.43T + 7T^{2} \) |
| 11 | \( 1 + 4.08T + 11T^{2} \) |
| 13 | \( 1 - 3.79T + 13T^{2} \) |
| 17 | \( 1 - 3.73T + 17T^{2} \) |
| 23 | \( 1 - 0.351T + 23T^{2} \) |
| 29 | \( 1 + 6.73T + 29T^{2} \) |
| 31 | \( 1 - 9.34T + 31T^{2} \) |
| 37 | \( 1 + 6.17T + 37T^{2} \) |
| 41 | \( 1 + 4.05T + 41T^{2} \) |
| 43 | \( 1 - 0.913T + 43T^{2} \) |
| 47 | \( 1 + 5.52T + 47T^{2} \) |
| 53 | \( 1 - 6.96T + 53T^{2} \) |
| 59 | \( 1 - 14.3T + 59T^{2} \) |
| 61 | \( 1 + 13.1T + 61T^{2} \) |
| 67 | \( 1 + 7.61T + 67T^{2} \) |
| 71 | \( 1 + 9.73T + 71T^{2} \) |
| 73 | \( 1 - 11.9T + 73T^{2} \) |
| 79 | \( 1 + 3.17T + 79T^{2} \) |
| 83 | \( 1 - 2.90T + 83T^{2} \) |
| 89 | \( 1 + 15.2T + 89T^{2} \) |
| 97 | \( 1 - 16.0T + 97T^{2} \) |
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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
−7.48682877721620524776972689356, −6.88722094024038690265319614710, −6.05593533523789091848240081030, −5.61828164047515370316790183900, −4.70670021956009945583389838143, −3.49715328518846138805582795368, −3.14874992714886443354421166552, −2.15049335620728939890177642662, −1.07775759960430977809533797187, 0,
1.07775759960430977809533797187, 2.15049335620728939890177642662, 3.14874992714886443354421166552, 3.49715328518846138805582795368, 4.70670021956009945583389838143, 5.61828164047515370316790183900, 6.05593533523789091848240081030, 6.88722094024038690265319614710, 7.48682877721620524776972689356