| L(s) = 1 | + 2-s + 4-s − 7-s + 8-s − 3·9-s + 4·11-s − 4·13-s − 14-s + 16-s + 4·17-s − 3·18-s + 4·19-s + 4·22-s + 23-s − 5·25-s − 4·26-s − 28-s + 2·31-s + 32-s + 4·34-s − 3·36-s + 10·37-s + 4·38-s − 4·41-s − 4·43-s + 4·44-s + 46-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.377·7-s + 0.353·8-s − 9-s + 1.20·11-s − 1.10·13-s − 0.267·14-s + 1/4·16-s + 0.970·17-s − 0.707·18-s + 0.917·19-s + 0.852·22-s + 0.208·23-s − 25-s − 0.784·26-s − 0.188·28-s + 0.359·31-s + 0.176·32-s + 0.685·34-s − 1/2·36-s + 1.64·37-s + 0.648·38-s − 0.624·41-s − 0.609·43-s + 0.603·44-s + 0.147·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 270802 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 270802 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 \) |
| good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 10 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
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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
−13.15424381443803, −12.36393125739370, −12.00477875696299, −11.74737045333800, −11.54586812295480, −10.70273019473275, −10.32092425663931, −9.674034899120313, −9.416199410528348, −8.965637077671546, −8.248561465857564, −7.744120081515280, −7.371638868509632, −6.847000941151948, −6.243519584640795, −5.811887504819426, −5.532583516600031, −4.805498487104513, −4.407165821671253, −3.684857043409039, −3.342170312457697, −2.738729379419118, −2.328503226187867, −1.461586759779826, −0.8858093167989513, 0,
0.8858093167989513, 1.461586759779826, 2.328503226187867, 2.738729379419118, 3.342170312457697, 3.684857043409039, 4.407165821671253, 4.805498487104513, 5.532583516600031, 5.811887504819426, 6.243519584640795, 6.847000941151948, 7.371638868509632, 7.744120081515280, 8.248561465857564, 8.965637077671546, 9.416199410528348, 9.674034899120313, 10.32092425663931, 10.70273019473275, 11.54586812295480, 11.74737045333800, 12.00477875696299, 12.36393125739370, 13.15424381443803
