| L(s) = 1 | − 0.709·2-s + 1.77·3-s − 0.497·4-s + 5-s − 1.25·6-s + 0.241·7-s + 1.06·8-s + 2.13·9-s − 0.709·10-s − 1.94·11-s − 0.880·12-s + 13-s − 0.170·14-s + 1.77·15-s − 0.255·16-s − 1.49·17-s − 1.51·18-s − 0.497·20-s + 0.426·21-s + 1.37·22-s + 1.13·23-s + 1.88·24-s + 25-s − 0.709·26-s + 2.01·27-s − 0.119·28-s − 1.25·30-s + ⋯ |
| L(s) = 1 | − 0.709·2-s + 1.77·3-s − 0.497·4-s + 5-s − 1.25·6-s + 0.241·7-s + 1.06·8-s + 2.13·9-s − 0.709·10-s − 1.94·11-s − 0.880·12-s + 13-s − 0.170·14-s + 1.77·15-s − 0.255·16-s − 1.49·17-s − 1.51·18-s − 0.497·20-s + 0.426·21-s + 1.37·22-s + 1.13·23-s + 1.88·24-s + 25-s − 0.709·26-s + 2.01·27-s − 0.119·28-s − 1.25·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.504290648\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.504290648\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 31 | \( 1 - T \) |
| good | 2 | \( 1 + 0.709T + T^{2} \) |
| 3 | \( 1 - 1.77T + T^{2} \) |
| 7 | \( 1 - 0.241T + T^{2} \) |
| 11 | \( 1 + 1.94T + T^{2} \) |
| 17 | \( 1 + 1.49T + T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 - 1.13T + T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 + 0.709T + T^{2} \) |
| 47 | \( 1 - 1.13T + T^{2} \) |
| 53 | \( 1 + 1.94T + T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 + 1.94T + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 - 0.241T + T^{2} \) |
| 97 | \( 1 - 1.77T + T^{2} \) |
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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
−9.044389748343957138003795940184, −8.740512232549285952948624930630, −8.065475235388433584300601395890, −7.45181641134705319280880790941, −6.41117599428878867782463155304, −5.07712297812359709301636419625, −4.47311772906868840213179001396, −3.16577836995136424423978186844, −2.41675961549851593372635937691, −1.48863415467420407122357934209,
1.48863415467420407122357934209, 2.41675961549851593372635937691, 3.16577836995136424423978186844, 4.47311772906868840213179001396, 5.07712297812359709301636419625, 6.41117599428878867782463155304, 7.45181641134705319280880790941, 8.065475235388433584300601395890, 8.740512232549285952948624930630, 9.044389748343957138003795940184
