| L(s) = 1 | + 24·7-s + 28·11-s + 74·13-s + 82·17-s + 92·19-s + 8·23-s + 138·29-s + 80·31-s − 30·37-s − 282·41-s − 4·43-s + 240·47-s + 233·49-s − 130·53-s − 596·59-s − 218·61-s + 436·67-s − 856·71-s + 998·73-s + 672·77-s − 32·79-s − 1.50e3·83-s + 246·89-s + 1.77e3·91-s − 866·97-s − 270·101-s + 1.49e3·103-s + ⋯ |
| L(s) = 1 | + 1.29·7-s + 0.767·11-s + 1.57·13-s + 1.16·17-s + 1.11·19-s + 0.0725·23-s + 0.883·29-s + 0.463·31-s − 0.133·37-s − 1.07·41-s − 0.0141·43-s + 0.744·47-s + 0.679·49-s − 0.336·53-s − 1.31·59-s − 0.457·61-s + 0.795·67-s − 1.43·71-s + 1.60·73-s + 0.994·77-s − 0.0455·79-s − 1.99·83-s + 0.292·89-s + 2.04·91-s − 0.906·97-s − 0.266·101-s + 1.43·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.610178533\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.610178533\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 24 T + p^{3} T^{2} \) |
| 11 | \( 1 - 28 T + p^{3} T^{2} \) |
| 13 | \( 1 - 74 T + p^{3} T^{2} \) |
| 17 | \( 1 - 82 T + p^{3} T^{2} \) |
| 19 | \( 1 - 92 T + p^{3} T^{2} \) |
| 23 | \( 1 - 8 T + p^{3} T^{2} \) |
| 29 | \( 1 - 138 T + p^{3} T^{2} \) |
| 31 | \( 1 - 80 T + p^{3} T^{2} \) |
| 37 | \( 1 + 30 T + p^{3} T^{2} \) |
| 41 | \( 1 + 282 T + p^{3} T^{2} \) |
| 43 | \( 1 + 4 T + p^{3} T^{2} \) |
| 47 | \( 1 - 240 T + p^{3} T^{2} \) |
| 53 | \( 1 + 130 T + p^{3} T^{2} \) |
| 59 | \( 1 + 596 T + p^{3} T^{2} \) |
| 61 | \( 1 + 218 T + p^{3} T^{2} \) |
| 67 | \( 1 - 436 T + p^{3} T^{2} \) |
| 71 | \( 1 + 856 T + p^{3} T^{2} \) |
| 73 | \( 1 - 998 T + p^{3} T^{2} \) |
| 79 | \( 1 + 32 T + p^{3} T^{2} \) |
| 83 | \( 1 + 1508 T + p^{3} T^{2} \) |
| 89 | \( 1 - 246 T + p^{3} T^{2} \) |
| 97 | \( 1 + 866 T + p^{3} T^{2} \) |
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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
−8.747140754500368337334338592119, −8.198097969563026750597942019560, −7.47420485954968219300071796335, −6.47425184975306120746819356963, −5.65661335678145225007903650797, −4.85762690659868484546026398669, −3.90514152228492132558566171605, −3.05847972843586962763972849065, −1.52132763296555907422716706388, −1.06578088634113477398216061899,
1.06578088634113477398216061899, 1.52132763296555907422716706388, 3.05847972843586962763972849065, 3.90514152228492132558566171605, 4.85762690659868484546026398669, 5.65661335678145225007903650797, 6.47425184975306120746819356963, 7.47420485954968219300071796335, 8.198097969563026750597942019560, 8.747140754500368337334338592119
