L(s) = 1 | + 0.381·5-s − i·7-s − 2.13i·11-s − 6.49i·13-s + 6.97i·17-s − 6.19·19-s + 1.82·23-s − 4.85·25-s + 0.694·29-s + 1.67i·31-s − 0.381i·35-s + 8.06i·37-s − 1.31i·41-s − 7.67·43-s − 6.82·47-s + ⋯ |
L(s) = 1 | + 0.170·5-s − 0.377i·7-s − 0.643i·11-s − 1.80i·13-s + 1.69i·17-s − 1.42·19-s + 0.380·23-s − 0.970·25-s + 0.129·29-s + 0.301i·31-s − 0.0644i·35-s + 1.32i·37-s − 0.204i·41-s − 1.17·43-s − 0.995·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.482 - 0.875i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.482 - 0.875i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5213128330\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5213128330\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + iT \) |
good | 5 | \( 1 - 0.381T + 5T^{2} \) |
| 11 | \( 1 + 2.13iT - 11T^{2} \) |
| 13 | \( 1 + 6.49iT - 13T^{2} \) |
| 17 | \( 1 - 6.97iT - 17T^{2} \) |
| 19 | \( 1 + 6.19T + 19T^{2} \) |
| 23 | \( 1 - 1.82T + 23T^{2} \) |
| 29 | \( 1 - 0.694T + 29T^{2} \) |
| 31 | \( 1 - 1.67iT - 31T^{2} \) |
| 37 | \( 1 - 8.06iT - 37T^{2} \) |
| 41 | \( 1 + 1.31iT - 41T^{2} \) |
| 43 | \( 1 + 7.67T + 43T^{2} \) |
| 47 | \( 1 + 6.82T + 47T^{2} \) |
| 53 | \( 1 + 0.954T + 53T^{2} \) |
| 59 | \( 1 + 12.8iT - 59T^{2} \) |
| 61 | \( 1 - 12.4iT - 61T^{2} \) |
| 67 | \( 1 + 0.634T + 67T^{2} \) |
| 71 | \( 1 + 5.79T + 71T^{2} \) |
| 73 | \( 1 - 8.13T + 73T^{2} \) |
| 79 | \( 1 - 14.1iT - 79T^{2} \) |
| 83 | \( 1 - 3.36iT - 83T^{2} \) |
| 89 | \( 1 - 15.9iT - 89T^{2} \) |
| 97 | \( 1 - 10.8T + 97T^{2} \) |
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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
−8.219332860774598142220758605323, −7.893179045362077662331265013535, −6.72070143145098319105329387874, −6.19919322601900481133559858933, −5.54216212979595990196522182303, −4.73311339060037879291431368018, −3.75059217019639796382523661284, −3.23122573554382312485318815340, −2.13636408269797885033680297265, −1.10618231408343695992888506152,
0.13264207384251671159826389810, 1.81154292538306704923872175775, 2.23805386499072799038433281840, 3.35194050381337937094250117482, 4.44130776527571590509558143265, 4.71746070402366046709730000552, 5.76570152367776408252251251782, 6.53432687492906609463583320476, 7.04223674713090738592598499251, 7.74911273861768013167613850952