| L(s) = 1 | + (−1.70 − 0.306i)3-s + 2.64i·7-s + (2.81 + 1.04i)9-s + 5.55i·11-s − 7.13i·13-s + 5.75·17-s + (0.811 − 4.51i)21-s + (−4.47 − 2.64i)27-s + 4.83i·29-s + (1.70 − 9.47i)33-s + (−2.18 + 12.1i)39-s − 1.28·47-s − 7.00·49-s + (−9.81 − 1.76i)51-s + (−2.76 + 7.43i)63-s + ⋯ |
| L(s) = 1 | + (−0.984 − 0.177i)3-s + 0.999i·7-s + (0.937 + 0.348i)9-s + 1.67i·11-s − 1.97i·13-s + 1.39·17-s + (0.177 − 0.984i)21-s + (−0.860 − 0.509i)27-s + 0.898i·29-s + (0.296 − 1.64i)33-s + (−0.350 + 1.94i)39-s − 0.187·47-s − 49-s + (−1.37 − 0.247i)51-s + (−0.348 + 0.937i)63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.177 - 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.177 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.097386398\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.097386398\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.70 + 0.306i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - 2.64iT \) |
| good | 11 | \( 1 - 5.55iT - 11T^{2} \) |
| 13 | \( 1 + 7.13iT - 13T^{2} \) |
| 17 | \( 1 - 5.75T + 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 - 4.83iT - 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 + 1.28T + 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 - 11.8iT - 71T^{2} \) |
| 73 | \( 1 - 10.5iT - 73T^{2} \) |
| 79 | \( 1 - 14.8T + 79T^{2} \) |
| 83 | \( 1 + 8.94T + 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 - 3.45iT - 97T^{2} \) |
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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
−9.549373413216956749271031349945, −8.326093129495892370570205325566, −7.64073910675438768378798799616, −6.96527369151216930599825704750, −5.90384044493065530968364186978, −5.36091923081255706460659200266, −4.78639220831935015287455588326, −3.45164049322575784038043900802, −2.34037757865854227351120497851, −1.13502930980800640525874500216,
0.52842630230909642203792953747, 1.58882374083613685118690580596, 3.35567775464401697311589236849, 4.06263002056889971002490285827, 4.88290246560736223539706181035, 5.91436650319971513093820900626, 6.42981786042464291816869868704, 7.25947820091254068166729946878, 8.048244382440632288711394151427, 9.090230669735015236218542814195
