| L(s) = 1 | − 2-s + 4-s + 4·7-s − 8-s + 4·11-s − 4·14-s + 16-s − 4·17-s − 4·19-s − 4·22-s − 4·23-s + 4·28-s + 2·29-s + 4·31-s − 32-s + 4·34-s + 12·37-s + 4·38-s − 12·41-s + 8·43-s + 4·44-s + 4·46-s + 9·49-s + 14·53-s − 4·56-s − 2·58-s − 2·59-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 1.51·7-s − 0.353·8-s + 1.20·11-s − 1.06·14-s + 1/4·16-s − 0.970·17-s − 0.917·19-s − 0.852·22-s − 0.834·23-s + 0.755·28-s + 0.371·29-s + 0.718·31-s − 0.176·32-s + 0.685·34-s + 1.97·37-s + 0.648·38-s − 1.87·41-s + 1.21·43-s + 0.603·44-s + 0.589·46-s + 9/7·49-s + 1.92·53-s − 0.534·56-s − 0.262·58-s − 0.260·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 167 | \( 1 - T \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 12 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 + 2 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−14.50790440594903, −13.82169698300480, −13.46082315885045, −12.73022248051213, −12.03359058486390, −11.69149485862450, −11.38786452232722, −10.78473249639909, −10.34314887854910, −9.784091832292535, −9.101567120068632, −8.647971776877771, −8.374792277756544, −7.742471295051042, −7.268625684037412, −6.482809467752777, −6.287592775398855, −5.490919962996515, −4.780022756757390, −4.140051155900834, −3.977728187068545, −2.685844546353415, −2.281338278246583, −1.505704890407836, −1.085608700864595, 0,
1.085608700864595, 1.505704890407836, 2.281338278246583, 2.685844546353415, 3.977728187068545, 4.140051155900834, 4.780022756757390, 5.490919962996515, 6.287592775398855, 6.482809467752777, 7.268625684037412, 7.742471295051042, 8.374792277756544, 8.647971776877771, 9.101567120068632, 9.784091832292535, 10.34314887854910, 10.78473249639909, 11.38786452232722, 11.69149485862450, 12.03359058486390, 12.73022248051213, 13.46082315885045, 13.82169698300480, 14.50790440594903
