| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 7-s − 8-s + 9-s + 12-s + 13-s + 14-s + 16-s + 3·17-s − 18-s − 5·19-s − 21-s − 23-s − 24-s − 5·25-s − 26-s + 27-s − 28-s − 9·29-s + 5·31-s − 32-s − 3·34-s + 36-s + 2·37-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.288·12-s + 0.277·13-s + 0.267·14-s + 1/4·16-s + 0.727·17-s − 0.235·18-s − 1.14·19-s − 0.218·21-s − 0.208·23-s − 0.204·24-s − 25-s − 0.196·26-s + 0.192·27-s − 0.188·28-s − 1.67·29-s + 0.898·31-s − 0.176·32-s − 0.514·34-s + 1/6·36-s + 0.328·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 116886 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 116886 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−13.79874955284284, −13.35583857745434, −12.89688665656316, −12.29455969522033, −11.99586598648521, −11.29320475855965, −10.74766932978984, −10.44711746826096, −9.739993312121955, −9.493003205102358, −8.897899637433212, −8.534517643262872, −7.825285492037044, −7.581369130525841, −7.062895556517878, −6.262361333294018, −5.983261079515614, −5.415227752242244, −4.504543356262086, −3.872609291692726, −3.606774702499376, −2.628142800989755, −2.342152901747529, −1.586763476538596, −0.8538505861213762, 0,
0.8538505861213762, 1.586763476538596, 2.342152901747529, 2.628142800989755, 3.606774702499376, 3.872609291692726, 4.504543356262086, 5.415227752242244, 5.983261079515614, 6.262361333294018, 7.062895556517878, 7.581369130525841, 7.825285492037044, 8.534517643262872, 8.897899637433212, 9.493003205102358, 9.739993312121955, 10.44711746826096, 10.74766932978984, 11.29320475855965, 11.99586598648521, 12.29455969522033, 12.89688665656316, 13.35583857745434, 13.79874955284284
