| L(s) = 1 | + 0.867·2-s − 0.246·4-s + 0.445·7-s − 1.08·8-s + 0.386·14-s − 0.692·16-s + 1.56·17-s + 1.24·19-s + 25-s − 0.109·28-s − 0.867·29-s + 1.80·31-s + 0.481·32-s + 1.35·34-s − 0.445·37-s + 1.08·38-s + 1.94·41-s − 1.80·43-s − 1.56·47-s − 0.801·49-s + 0.867·50-s − 1.56·53-s − 0.481·56-s − 0.753·58-s + 1.56·62-s + 1.10·64-s − 0.386·68-s + ⋯ |
| L(s) = 1 | + 0.867·2-s − 0.246·4-s + 0.445·7-s − 1.08·8-s + 0.386·14-s − 0.692·16-s + 1.56·17-s + 1.24·19-s + 25-s − 0.109·28-s − 0.867·29-s + 1.80·31-s + 0.481·32-s + 1.35·34-s − 0.445·37-s + 1.08·38-s + 1.94·41-s − 1.80·43-s − 1.56·47-s − 0.801·49-s + 0.867·50-s − 1.56·53-s − 0.481·56-s − 0.753·58-s + 1.56·62-s + 1.10·64-s − 0.386·68-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.667030556\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.667030556\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 223 | \( 1 + T \) |
| good | 2 | \( 1 - 0.867T + T^{2} \) |
| 5 | \( 1 - T^{2} \) |
| 7 | \( 1 - 0.445T + T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 - 1.56T + T^{2} \) |
| 19 | \( 1 - 1.24T + T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 + 0.867T + T^{2} \) |
| 31 | \( 1 - 1.80T + T^{2} \) |
| 37 | \( 1 + 0.445T + T^{2} \) |
| 41 | \( 1 - 1.94T + T^{2} \) |
| 43 | \( 1 + 1.80T + T^{2} \) |
| 47 | \( 1 + 1.56T + T^{2} \) |
| 53 | \( 1 + 1.56T + T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - 1.80T + T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + 0.867T + T^{2} \) |
| 89 | \( 1 + 1.94T + T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−9.605051685868398615074046061972, −8.391070486839570819765593461502, −7.902583248283845500419789154937, −6.84308060085814779738380291134, −5.93137052386459435899812909862, −5.16980174853295639128473919953, −4.64715203508313680488517615062, −3.49312396831364028628472554459, −2.92347842461185816926138548066, −1.26116329680899468838070771643,
1.26116329680899468838070771643, 2.92347842461185816926138548066, 3.49312396831364028628472554459, 4.64715203508313680488517615062, 5.16980174853295639128473919953, 5.93137052386459435899812909862, 6.84308060085814779738380291134, 7.902583248283845500419789154937, 8.391070486839570819765593461502, 9.605051685868398615074046061972
