| L(s) = 1 | − 1.49·2-s + 1.13·3-s + 1.24·4-s + 5-s − 1.70·6-s − 1.94·7-s − 0.360·8-s + 0.290·9-s − 1.49·10-s + 1.77·11-s + 1.41·12-s + 13-s + 2.90·14-s + 1.13·15-s − 0.700·16-s + 0.241·17-s − 0.435·18-s + 1.24·20-s − 2.20·21-s − 2.65·22-s − 0.709·23-s − 0.410·24-s + 25-s − 1.49·26-s − 0.805·27-s − 2.41·28-s − 1.70·30-s + ⋯ |
| L(s) = 1 | − 1.49·2-s + 1.13·3-s + 1.24·4-s + 5-s − 1.70·6-s − 1.94·7-s − 0.360·8-s + 0.290·9-s − 1.49·10-s + 1.77·11-s + 1.41·12-s + 13-s + 2.90·14-s + 1.13·15-s − 0.700·16-s + 0.241·17-s − 0.435·18-s + 1.24·20-s − 2.20·21-s − 2.65·22-s − 0.709·23-s − 0.410·24-s + 25-s − 1.49·26-s − 0.805·27-s − 2.41·28-s − 1.70·30-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
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8897987457\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8897987457\) |
| \(L(1)\) |
|
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 | \( 1 + 1.49T + T^{2} \) |
| 3 | \( 1 - 1.13T + T^{2} \) |
| 7 | \( 1 + 1.94T + T^{2} \) |
| 11 | \( 1 - 1.77T + T^{2} \) |
| 17 | \( 1 - 0.241T + T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + 0.709T + T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 + 1.49T + T^{2} \) |
| 47 | \( 1 + 0.709T + T^{2} \) |
| 53 | \( 1 - 1.77T + T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - 1.77T + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + 1.94T + T^{2} \) |
| 97 | \( 1 - 1.13T + T^{2} \) |
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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
−9.298043484344438941052163472721, −8.766262899806152035433056359161, −8.265686471172096623418016105361, −6.93832486220460781318280558304, −6.57049811885079461933199549599, −5.86356680986169189328160252009, −3.97276575357474029598654748630, −3.24675194472769517197183161774, −2.26357805456589487590510749603, −1.19300894834485647613845907005,
1.19300894834485647613845907005, 2.26357805456589487590510749603, 3.24675194472769517197183161774, 3.97276575357474029598654748630, 5.86356680986169189328160252009, 6.57049811885079461933199549599, 6.93832486220460781318280558304, 8.265686471172096623418016105361, 8.766262899806152035433056359161, 9.298043484344438941052163472721
