| L(s) = 1 | + (0.225 − 1.71i)3-s − i·5-s + i·7-s + (−2.89 − 0.774i)9-s − 2.63·11-s − 2.35·13-s + (−1.71 − 0.225i)15-s + 5.43i·17-s + 3.16i·19-s + (1.71 + 0.225i)21-s − 2.36·23-s − 25-s + (−1.98 + 4.80i)27-s + 9.68i·29-s − 4.06i·31-s + ⋯ |
| L(s) = 1 | + (0.130 − 0.991i)3-s − 0.447i·5-s + 0.377i·7-s + (−0.966 − 0.258i)9-s − 0.793·11-s − 0.652·13-s + (−0.443 − 0.0582i)15-s + 1.31i·17-s + 0.726i·19-s + (0.374 + 0.0492i)21-s − 0.492·23-s − 0.200·25-s + (−0.381 + 0.924i)27-s + 1.79i·29-s − 0.730i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.130 - 0.991i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.130 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6039950037\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6039950037\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.225 + 1.71i)T \) |
| 5 | \( 1 + iT \) |
| 7 | \( 1 - iT \) |
| good | 11 | \( 1 + 2.63T + 11T^{2} \) |
| 13 | \( 1 + 2.35T + 13T^{2} \) |
| 17 | \( 1 - 5.43iT - 17T^{2} \) |
| 19 | \( 1 - 3.16iT - 19T^{2} \) |
| 23 | \( 1 + 2.36T + 23T^{2} \) |
| 29 | \( 1 - 9.68iT - 29T^{2} \) |
| 31 | \( 1 + 4.06iT - 31T^{2} \) |
| 37 | \( 1 - 9.77T + 37T^{2} \) |
| 41 | \( 1 + 0.361iT - 41T^{2} \) |
| 43 | \( 1 + 2.36iT - 43T^{2} \) |
| 47 | \( 1 + 12.4T + 47T^{2} \) |
| 53 | \( 1 - 9.67iT - 53T^{2} \) |
| 59 | \( 1 - 5.96T + 59T^{2} \) |
| 61 | \( 1 + 1.45T + 61T^{2} \) |
| 67 | \( 1 + 13.9iT - 67T^{2} \) |
| 71 | \( 1 + 14.7T + 71T^{2} \) |
| 73 | \( 1 + 1.16T + 73T^{2} \) |
| 79 | \( 1 - 8.78iT - 79T^{2} \) |
| 83 | \( 1 + 2.37T + 83T^{2} \) |
| 89 | \( 1 + 7.06iT - 89T^{2} \) |
| 97 | \( 1 + 4.74T + 97T^{2} \) |
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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
−9.403981413683791213540387833584, −8.508198296427628963141953354029, −7.993450126540943707780040859124, −7.30832205715515958212129695039, −6.21531319277439183721160198033, −5.67621609494935205918355604829, −4.69314936965364383258278683658, −3.43044102324585423438656799523, −2.36619533110570051623671617016, −1.43665485781503178756529250359,
0.21655119866969495107375127509, 2.45311537108973060760689812974, 3.04926411721459907603962033707, 4.26928945562676919009551531410, 4.89132731753087569417725154619, 5.77307941734128839221860610103, 6.82077897296214023828913343732, 7.68877289119997446349371678593, 8.347097693798992191012384052373, 9.516871736074502877337122219151
