L(s) = 1 | − 1.23·2-s + 0.767·3-s − 0.485·4-s − 0.944·6-s + 7-s + 3.05·8-s − 2.41·9-s + 2.89·11-s − 0.372·12-s − 5.98·13-s − 1.23·14-s − 2.79·16-s + 3.74·17-s + 2.96·18-s + 3.96·19-s + 0.767·21-s − 3.55·22-s − 23-s + 2.34·24-s + 7.37·26-s − 4.15·27-s − 0.485·28-s − 5.11·29-s − 2.68·31-s − 2.67·32-s + 2.21·33-s − 4.60·34-s + ⋯ |
L(s) = 1 | − 0.870·2-s + 0.443·3-s − 0.242·4-s − 0.385·6-s + 0.377·7-s + 1.08·8-s − 0.803·9-s + 0.871·11-s − 0.107·12-s − 1.66·13-s − 0.328·14-s − 0.698·16-s + 0.908·17-s + 0.699·18-s + 0.910·19-s + 0.167·21-s − 0.758·22-s − 0.208·23-s + 0.479·24-s + 1.44·26-s − 0.799·27-s − 0.0916·28-s − 0.950·29-s − 0.483·31-s − 0.473·32-s + 0.386·33-s − 0.790·34-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 | \( 1 + 1.23T + 2T^{2} \) |
| 3 | \( 1 - 0.767T + 3T^{2} \) |
| 11 | \( 1 - 2.89T + 11T^{2} \) |
| 13 | \( 1 + 5.98T + 13T^{2} \) |
| 17 | \( 1 - 3.74T + 17T^{2} \) |
| 19 | \( 1 - 3.96T + 19T^{2} \) |
| 29 | \( 1 + 5.11T + 29T^{2} \) |
| 31 | \( 1 + 2.68T + 31T^{2} \) |
| 37 | \( 1 + 0.781T + 37T^{2} \) |
| 41 | \( 1 + 3.75T + 41T^{2} \) |
| 43 | \( 1 - 9.64T + 43T^{2} \) |
| 47 | \( 1 + 6.15T + 47T^{2} \) |
| 53 | \( 1 - 2.86T + 53T^{2} \) |
| 59 | \( 1 - 5.12T + 59T^{2} \) |
| 61 | \( 1 + 1.72T + 61T^{2} \) |
| 67 | \( 1 + 6.58T + 67T^{2} \) |
| 71 | \( 1 + 13.5T + 71T^{2} \) |
| 73 | \( 1 - 15.6T + 73T^{2} \) |
| 79 | \( 1 + 3.72T + 79T^{2} \) |
| 83 | \( 1 + 0.558T + 83T^{2} \) |
| 89 | \( 1 + 8.00T + 89T^{2} \) |
| 97 | \( 1 + 12.4T + 97T^{2} \) |
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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
−8.122282424413925116469114404370, −7.54846792559978849773431449603, −7.04714086399916860100062843476, −5.71391558245094995972871299435, −5.15232237720313680787169183333, −4.22010781422947404061207061033, −3.33431063307611131662088100679, −2.29525042565562279786278973031, −1.29433341776264960411372516570, 0,
1.29433341776264960411372516570, 2.29525042565562279786278973031, 3.33431063307611131662088100679, 4.22010781422947404061207061033, 5.15232237720313680787169183333, 5.71391558245094995972871299435, 7.04714086399916860100062843476, 7.54846792559978849773431449603, 8.122282424413925116469114404370