| L(s) = 1 | − 2.53·2-s + 3-s + 4.43·4-s − 2.53·6-s − 1.04·7-s − 6.19·8-s + 9-s + 2.97·11-s + 4.43·12-s − 5.66·13-s + 2.64·14-s + 6.83·16-s + 5.08·17-s − 2.53·18-s − 5.37·19-s − 1.04·21-s − 7.53·22-s − 3.86·23-s − 6.19·24-s + 14.3·26-s + 27-s − 4.61·28-s + 0.679·29-s + 0.850·31-s − 4.95·32-s + 2.97·33-s − 12.9·34-s + ⋯ |
| L(s) = 1 | − 1.79·2-s + 0.577·3-s + 2.21·4-s − 1.03·6-s − 0.393·7-s − 2.18·8-s + 0.333·9-s + 0.895·11-s + 1.28·12-s − 1.57·13-s + 0.705·14-s + 1.70·16-s + 1.23·17-s − 0.598·18-s − 1.23·19-s − 0.227·21-s − 1.60·22-s − 0.804·23-s − 1.26·24-s + 2.82·26-s + 0.192·27-s − 0.873·28-s + 0.126·29-s + 0.152·31-s − 0.875·32-s + 0.517·33-s − 2.21·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + 2.53T + 2T^{2} \) |
| 7 | \( 1 + 1.04T + 7T^{2} \) |
| 11 | \( 1 - 2.97T + 11T^{2} \) |
| 13 | \( 1 + 5.66T + 13T^{2} \) |
| 17 | \( 1 - 5.08T + 17T^{2} \) |
| 19 | \( 1 + 5.37T + 19T^{2} \) |
| 23 | \( 1 + 3.86T + 23T^{2} \) |
| 29 | \( 1 - 0.679T + 29T^{2} \) |
| 31 | \( 1 - 0.850T + 31T^{2} \) |
| 37 | \( 1 - 1.61T + 37T^{2} \) |
| 41 | \( 1 - 1.16T + 41T^{2} \) |
| 43 | \( 1 + 5.68T + 43T^{2} \) |
| 47 | \( 1 + 3.28T + 47T^{2} \) |
| 53 | \( 1 + 12.6T + 53T^{2} \) |
| 59 | \( 1 - 3.21T + 59T^{2} \) |
| 61 | \( 1 + 5.42T + 61T^{2} \) |
| 67 | \( 1 - 0.929T + 67T^{2} \) |
| 71 | \( 1 + 1.41T + 71T^{2} \) |
| 73 | \( 1 + 11.3T + 73T^{2} \) |
| 79 | \( 1 + 1.44T + 79T^{2} \) |
| 83 | \( 1 - 11.4T + 83T^{2} \) |
| 89 | \( 1 - 9.07T + 89T^{2} \) |
| 97 | \( 1 - 6.02T + 97T^{2} \) |
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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
−8.929840522117111138716198782625, −8.094118662702940609756321038432, −7.62262272314330601234222395762, −6.76427659605116403525798639190, −6.14201108728692152788416009006, −4.67666427289859093325953742505, −3.39558079115476299675619020344, −2.41168519262975522201456685623, −1.49268485536215712920412218039, 0,
1.49268485536215712920412218039, 2.41168519262975522201456685623, 3.39558079115476299675619020344, 4.67666427289859093325953742505, 6.14201108728692152788416009006, 6.76427659605116403525798639190, 7.62262272314330601234222395762, 8.094118662702940609756321038432, 8.929840522117111138716198782625
