| L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s + 2.88·7-s − 8-s + 9-s + 10-s − 4.89·11-s − 12-s + 3.40·13-s − 2.88·14-s + 15-s + 16-s − 18-s + 3.07·19-s − 20-s − 2.88·21-s + 4.89·22-s − 3.23·23-s + 24-s + 25-s − 3.40·26-s − 27-s + 2.88·28-s + 3.58·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.447·5-s + 0.408·6-s + 1.09·7-s − 0.353·8-s + 0.333·9-s + 0.316·10-s − 1.47·11-s − 0.288·12-s + 0.945·13-s − 0.772·14-s + 0.258·15-s + 0.250·16-s − 0.235·18-s + 0.706·19-s − 0.223·20-s − 0.630·21-s + 1.04·22-s − 0.674·23-s + 0.204·24-s + 0.200·25-s − 0.668·26-s − 0.192·27-s + 0.546·28-s + 0.666·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8670 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8670 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 17 | \( 1 \) |
| good | 7 | \( 1 - 2.88T + 7T^{2} \) |
| 11 | \( 1 + 4.89T + 11T^{2} \) |
| 13 | \( 1 - 3.40T + 13T^{2} \) |
| 19 | \( 1 - 3.07T + 19T^{2} \) |
| 23 | \( 1 + 3.23T + 23T^{2} \) |
| 29 | \( 1 - 3.58T + 29T^{2} \) |
| 31 | \( 1 + 1.70T + 31T^{2} \) |
| 37 | \( 1 + 5.96T + 37T^{2} \) |
| 41 | \( 1 - 0.855T + 41T^{2} \) |
| 43 | \( 1 + 8.25T + 43T^{2} \) |
| 47 | \( 1 - 10.5T + 47T^{2} \) |
| 53 | \( 1 + 6.32T + 53T^{2} \) |
| 59 | \( 1 + 4.19T + 59T^{2} \) |
| 61 | \( 1 - 6.95T + 61T^{2} \) |
| 67 | \( 1 - 12.7T + 67T^{2} \) |
| 71 | \( 1 + 13.9T + 71T^{2} \) |
| 73 | \( 1 + 16.6T + 73T^{2} \) |
| 79 | \( 1 - 12.2T + 79T^{2} \) |
| 83 | \( 1 - 9.05T + 83T^{2} \) |
| 89 | \( 1 - 0.114T + 89T^{2} \) |
| 97 | \( 1 - 3.30T + 97T^{2} \) |
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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
−7.56925107082401433193536691180, −6.93319693168523675424867066613, −6.03018222752819929286616885981, −5.35001973071544545715356183915, −4.82295594766885497051130706597, −3.87708326013404391903214296561, −2.96205317957292451067615144146, −1.95542837727501456118466883675, −1.10957773765156548150229456814, 0,
1.10957773765156548150229456814, 1.95542837727501456118466883675, 2.96205317957292451067615144146, 3.87708326013404391903214296561, 4.82295594766885497051130706597, 5.35001973071544545715356183915, 6.03018222752819929286616885981, 6.93319693168523675424867066613, 7.56925107082401433193536691180
