| L(s) = 1 | + 3-s + 5·7-s + 9-s − 6·11-s + 3·13-s − 2·17-s + 19-s + 5·21-s + 2·23-s + 27-s − 6·29-s − 3·31-s − 6·33-s + 6·37-s + 3·39-s + 4·41-s + 11·43-s + 10·47-s + 18·49-s − 2·51-s + 8·53-s + 57-s − 6·59-s − 3·61-s + 5·63-s − 67-s + 2·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.88·7-s + 1/3·9-s − 1.80·11-s + 0.832·13-s − 0.485·17-s + 0.229·19-s + 1.09·21-s + 0.417·23-s + 0.192·27-s − 1.11·29-s − 0.538·31-s − 1.04·33-s + 0.986·37-s + 0.480·39-s + 0.624·41-s + 1.67·43-s + 1.45·47-s + 18/7·49-s − 0.280·51-s + 1.09·53-s + 0.132·57-s − 0.781·59-s − 0.384·61-s + 0.629·63-s − 0.122·67-s + 0.240·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.984017143\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.984017143\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 5 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 - 11 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 - 8 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 3 T + p T^{2} \) |
| 67 | \( 1 + T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 16 T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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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
−8.201863862967182335757742038887, −7.64891828338497925500545731242, −7.26308810477019149067741484925, −5.83812866594127097306255052917, −5.34832608585209405533922601574, −4.55446701848362328075683363765, −3.86642697921038195751338100321, −2.62268374317419374174264080329, −2.08508034755959974522162787938, −0.962489928612260611078268480682,
0.962489928612260611078268480682, 2.08508034755959974522162787938, 2.62268374317419374174264080329, 3.86642697921038195751338100321, 4.55446701848362328075683363765, 5.34832608585209405533922601574, 5.83812866594127097306255052917, 7.26308810477019149067741484925, 7.64891828338497925500545731242, 8.201863862967182335757742038887
