| L(s) = 1 | + 2.04·2-s + 1.31·3-s + 2.16·4-s + 2.68·6-s − 7-s + 0.333·8-s − 1.27·9-s + 0.520·11-s + 2.84·12-s − 6.55·13-s − 2.04·14-s − 3.64·16-s − 0.414·17-s − 2.60·18-s − 5.89·19-s − 1.31·21-s + 1.06·22-s − 23-s + 0.438·24-s − 13.3·26-s − 5.61·27-s − 2.16·28-s + 9.15·29-s + 4.58·31-s − 8.10·32-s + 0.684·33-s − 0.845·34-s + ⋯ |
| L(s) = 1 | + 1.44·2-s + 0.758·3-s + 1.08·4-s + 1.09·6-s − 0.377·7-s + 0.118·8-s − 0.424·9-s + 0.157·11-s + 0.820·12-s − 1.81·13-s − 0.545·14-s − 0.911·16-s − 0.100·17-s − 0.613·18-s − 1.35·19-s − 0.286·21-s + 0.226·22-s − 0.208·23-s + 0.0895·24-s − 2.62·26-s − 1.08·27-s − 0.408·28-s + 1.70·29-s + 0.823·31-s − 1.43·32-s + 0.119·33-s − 0.145·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 2 | \( 1 - 2.04T + 2T^{2} \) |
| 3 | \( 1 - 1.31T + 3T^{2} \) |
| 11 | \( 1 - 0.520T + 11T^{2} \) |
| 13 | \( 1 + 6.55T + 13T^{2} \) |
| 17 | \( 1 + 0.414T + 17T^{2} \) |
| 19 | \( 1 + 5.89T + 19T^{2} \) |
| 29 | \( 1 - 9.15T + 29T^{2} \) |
| 31 | \( 1 - 4.58T + 31T^{2} \) |
| 37 | \( 1 - 4.71T + 37T^{2} \) |
| 41 | \( 1 + 11.5T + 41T^{2} \) |
| 43 | \( 1 + 7.31T + 43T^{2} \) |
| 47 | \( 1 - 1.49T + 47T^{2} \) |
| 53 | \( 1 + 7.40T + 53T^{2} \) |
| 59 | \( 1 - 2.78T + 59T^{2} \) |
| 61 | \( 1 - 10.5T + 61T^{2} \) |
| 67 | \( 1 + 3.58T + 67T^{2} \) |
| 71 | \( 1 - 7.82T + 71T^{2} \) |
| 73 | \( 1 + 1.04T + 73T^{2} \) |
| 79 | \( 1 + 9.97T + 79T^{2} \) |
| 83 | \( 1 - 6.07T + 83T^{2} \) |
| 89 | \( 1 + 16.4T + 89T^{2} \) |
| 97 | \( 1 - 1.87T + 97T^{2} \) |
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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
−8.215791872368576198305007501902, −7.00223660105949767974597365665, −6.57862931177251643447094099756, −5.72133901414830056899806124803, −4.83649505895891017259608912242, −4.37111562051659733611917709497, −3.36628835520234064836863410012, −2.71710630523569690853305408715, −2.13702664669941800506845113228, 0,
2.13702664669941800506845113228, 2.71710630523569690853305408715, 3.36628835520234064836863410012, 4.37111562051659733611917709497, 4.83649505895891017259608912242, 5.72133901414830056899806124803, 6.57862931177251643447094099756, 7.00223660105949767974597365665, 8.215791872368576198305007501902
