L(s) = 1 | + 3·3-s + 2·5-s − 7-s + 6·9-s + 5·11-s + 2·13-s + 6·15-s − 3·21-s + 2·23-s − 25-s + 9·27-s − 6·29-s − 4·31-s + 15·33-s − 2·35-s + 37-s + 6·39-s − 9·41-s − 2·43-s + 12·45-s − 9·47-s − 6·49-s − 53-s + 10·55-s − 8·59-s + 8·61-s − 6·63-s + ⋯ |
L(s) = 1 | + 1.73·3-s + 0.894·5-s − 0.377·7-s + 2·9-s + 1.50·11-s + 0.554·13-s + 1.54·15-s − 0.654·21-s + 0.417·23-s − 1/5·25-s + 1.73·27-s − 1.11·29-s − 0.718·31-s + 2.61·33-s − 0.338·35-s + 0.164·37-s + 0.960·39-s − 1.40·41-s − 0.304·43-s + 1.78·45-s − 1.31·47-s − 6/7·49-s − 0.137·53-s + 1.34·55-s − 1.04·59-s + 1.02·61-s − 0.755·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.233389815\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.233389815\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 37 | \( 1 - T \) |
good | 3 | \( 1 - p T + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 + T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 9 T + p T^{2} \) |
| 73 | \( 1 + T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 15 T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 97 | \( 1 - 4 T + p T^{2} \) |
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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
−9.060335490878057293168547796836, −8.431027512534208902956816675291, −7.54531634893152851657360813522, −6.70774001599674103257567801405, −6.09322645396956246911497609886, −4.85099413143575983714565439499, −3.66327550371300139045999952635, −3.38130050165522023852320334691, −2.07987843940684591281612159053, −1.49555061080500033055100672343,
1.49555061080500033055100672343, 2.07987843940684591281612159053, 3.38130050165522023852320334691, 3.66327550371300139045999952635, 4.85099413143575983714565439499, 6.09322645396956246911497609886, 6.70774001599674103257567801405, 7.54531634893152851657360813522, 8.431027512534208902956816675291, 9.060335490878057293168547796836