| L(s) = 1 | − 3·3-s − 4·7-s + 9·9-s − 40·11-s + 90·13-s + 70·17-s − 40·19-s + 12·21-s + 108·23-s − 27·27-s + 166·29-s + 40·31-s + 120·33-s + 130·37-s − 270·39-s − 310·41-s − 268·43-s − 556·47-s − 327·49-s − 210·51-s + 370·53-s + 120·57-s − 240·59-s − 130·61-s − 36·63-s + 876·67-s − 324·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.215·7-s + 1/3·9-s − 1.09·11-s + 1.92·13-s + 0.998·17-s − 0.482·19-s + 0.124·21-s + 0.979·23-s − 0.192·27-s + 1.06·29-s + 0.231·31-s + 0.633·33-s + 0.577·37-s − 1.10·39-s − 1.18·41-s − 0.950·43-s − 1.72·47-s − 0.953·49-s − 0.576·51-s + 0.958·53-s + 0.278·57-s − 0.529·59-s − 0.272·61-s − 0.0719·63-s + 1.59·67-s − 0.565·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.757740882\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.757740882\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + p T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 4 T + p^{3} T^{2} \) |
| 11 | \( 1 + 40 T + p^{3} T^{2} \) |
| 13 | \( 1 - 90 T + p^{3} T^{2} \) |
| 17 | \( 1 - 70 T + p^{3} T^{2} \) |
| 19 | \( 1 + 40 T + p^{3} T^{2} \) |
| 23 | \( 1 - 108 T + p^{3} T^{2} \) |
| 29 | \( 1 - 166 T + p^{3} T^{2} \) |
| 31 | \( 1 - 40 T + p^{3} T^{2} \) |
| 37 | \( 1 - 130 T + p^{3} T^{2} \) |
| 41 | \( 1 + 310 T + p^{3} T^{2} \) |
| 43 | \( 1 + 268 T + p^{3} T^{2} \) |
| 47 | \( 1 + 556 T + p^{3} T^{2} \) |
| 53 | \( 1 - 370 T + p^{3} T^{2} \) |
| 59 | \( 1 + 240 T + p^{3} T^{2} \) |
| 61 | \( 1 + 130 T + p^{3} T^{2} \) |
| 67 | \( 1 - 876 T + p^{3} T^{2} \) |
| 71 | \( 1 - 840 T + p^{3} T^{2} \) |
| 73 | \( 1 + 250 T + p^{3} T^{2} \) |
| 79 | \( 1 - 880 T + p^{3} T^{2} \) |
| 83 | \( 1 + 188 T + p^{3} T^{2} \) |
| 89 | \( 1 + 726 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1550 T + p^{3} T^{2} \) |
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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
−8.282375215826227248500635563202, −8.156934997616011974596503727131, −6.82656169215741030924238181994, −6.34797418825491860358263670327, −5.45246309958657083386664968130, −4.83144357549679585456733578333, −3.67047829184532088491135846759, −2.96009649854331603670437710199, −1.56781777610032261282560141714, −0.63776740385633488303951328186,
0.63776740385633488303951328186, 1.56781777610032261282560141714, 2.96009649854331603670437710199, 3.67047829184532088491135846759, 4.83144357549679585456733578333, 5.45246309958657083386664968130, 6.34797418825491860358263670327, 6.82656169215741030924238181994, 8.156934997616011974596503727131, 8.282375215826227248500635563202
