| L(s) = 1 | + 2-s − 3-s + 4-s − 6-s + 2i·7-s + 8-s − 2·9-s − 5i·11-s − 12-s − i·13-s + 2i·14-s + 16-s − 2·17-s − 2·18-s − 4i·19-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.408·6-s + 0.755i·7-s + 0.353·8-s − 0.666·9-s − 1.50i·11-s − 0.288·12-s − 0.277i·13-s + 0.534i·14-s + 0.250·16-s − 0.485·17-s − 0.471·18-s − 0.917i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0830 + 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0830 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.526420793\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.526420793\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 \) |
| 29 | \( 1 + (5 + 2i)T \) |
| good | 3 | \( 1 + T + 3T^{2} \) |
| 7 | \( 1 - 2iT - 7T^{2} \) |
| 11 | \( 1 + 5iT - 11T^{2} \) |
| 13 | \( 1 + iT - 13T^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 19 | \( 1 + 4iT - 19T^{2} \) |
| 23 | \( 1 + 6iT - 23T^{2} \) |
| 31 | \( 1 - 5iT - 31T^{2} \) |
| 37 | \( 1 - 8T + 37T^{2} \) |
| 41 | \( 1 + 10iT - 41T^{2} \) |
| 43 | \( 1 - 9T + 43T^{2} \) |
| 47 | \( 1 - 3T + 47T^{2} \) |
| 53 | \( 1 + iT - 53T^{2} \) |
| 59 | \( 1 + 10T + 59T^{2} \) |
| 61 | \( 1 + 10iT - 61T^{2} \) |
| 67 | \( 1 + 8iT - 67T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 + 16T + 73T^{2} \) |
| 79 | \( 1 - iT - 79T^{2} \) |
| 83 | \( 1 - 14iT - 83T^{2} \) |
| 89 | \( 1 + 14iT - 89T^{2} \) |
| 97 | \( 1 + 2T + 97T^{2} \) |
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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
−9.035081040473943974482121580754, −8.712886386613087680065536257966, −7.64455995498451515650231633919, −6.48868778056730537354213718191, −5.91780388695449228495106273879, −5.35296248584864309524614005839, −4.36541657079840703113616755052, −3.13473793645632531580293932047, −2.42779227717659701405065795187, −0.51371193032637462473491901349,
1.48329155079401262386443180218, 2.71732805509541168403481958684, 4.02998248938515837541060258883, 4.55002360444251005213259639880, 5.64209652715543392642684476361, 6.22186394519925670254467387333, 7.35378838434309006612437374240, 7.64087204789015925113913405095, 9.052896965761173809508087689577, 9.867570504594026311819857712780
