| L(s) = 1 | + (2.11 − 0.726i)5-s − 4.05i·7-s + 0.985i·11-s + 4.94·13-s + 4.52i·17-s − 2.60i·19-s + 3.53i·23-s + (3.94 − 3.07i)25-s − 7.59i·29-s + 3.28·31-s + (−2.94 − 8.58i)35-s − 0.945·37-s − 0.568·41-s − 8.45·43-s − 2.60i·47-s + ⋯ |
| L(s) = 1 | + (0.945 − 0.324i)5-s − 1.53i·7-s + 0.297i·11-s + 1.37·13-s + 1.09i·17-s − 0.597i·19-s + 0.737i·23-s + (0.789 − 0.614i)25-s − 1.41i·29-s + 0.589·31-s + (−0.497 − 1.45i)35-s − 0.155·37-s − 0.0887·41-s − 1.29·43-s − 0.379i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.530 + 0.847i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.530 + 0.847i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.103395712\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.103395712\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-2.11 + 0.726i)T \) |
| good | 7 | \( 1 + 4.05iT - 7T^{2} \) |
| 11 | \( 1 - 0.985iT - 11T^{2} \) |
| 13 | \( 1 - 4.94T + 13T^{2} \) |
| 17 | \( 1 - 4.52iT - 17T^{2} \) |
| 19 | \( 1 + 2.60iT - 19T^{2} \) |
| 23 | \( 1 - 3.53iT - 23T^{2} \) |
| 29 | \( 1 + 7.59iT - 29T^{2} \) |
| 31 | \( 1 - 3.28T + 31T^{2} \) |
| 37 | \( 1 + 0.945T + 37T^{2} \) |
| 41 | \( 1 + 0.568T + 41T^{2} \) |
| 43 | \( 1 + 8.45T + 43T^{2} \) |
| 47 | \( 1 + 2.60iT - 47T^{2} \) |
| 53 | \( 1 + 0.229T + 53T^{2} \) |
| 59 | \( 1 + 9.10iT - 59T^{2} \) |
| 61 | \( 1 + 11.0iT - 61T^{2} \) |
| 67 | \( 1 - 8.45T + 67T^{2} \) |
| 71 | \( 1 - 1.43T + 71T^{2} \) |
| 73 | \( 1 - 11.9iT - 73T^{2} \) |
| 79 | \( 1 + 3.28T + 79T^{2} \) |
| 83 | \( 1 - 9.89T + 83T^{2} \) |
| 89 | \( 1 + 12.3T + 89T^{2} \) |
| 97 | \( 1 - 3.23iT - 97T^{2} \) |
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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
−9.609721465798060147324964145280, −8.520945424676865571711541100269, −7.914384079456992713071678653784, −6.75802313770496469729510105736, −6.28524417103142501143136642842, −5.23008547850615964295876856968, −4.22774655341874005962782919726, −3.49005920146026080026101974054, −1.92390376170790479077590070880, −0.942590635634281089457906347394,
1.46064666476289788937394305604, 2.58190790307188608513922138075, 3.33381591635707378793982800884, 4.86680919672943546384935410365, 5.68144034145938091198566397057, 6.20191611573038067994760347136, 7.04374985726521084231804453842, 8.427013298908508975684152539131, 8.787691200042909579021100951798, 9.558838355286580067803828837126
