| L(s) = 1 | + (−1 + i)2-s + 2.44i·3-s − 2i·4-s + (−2.44 − 2.44i)6-s + (−1 − 2.44i)7-s + (2 + 2i)8-s − 2.99·9-s − 5i·11-s + 4.89·12-s + 2.44·13-s + (3.44 + 1.44i)14-s − 4·16-s + 4.89·17-s + (2.99 − 2.99i)18-s + (5.99 − 2.44i)21-s + (5 + 5i)22-s + ⋯ |
| L(s) = 1 | + (−0.707 + 0.707i)2-s + 1.41i·3-s − i·4-s + (−0.999 − 0.999i)6-s + (−0.377 − 0.925i)7-s + (0.707 + 0.707i)8-s − 0.999·9-s − 1.50i·11-s + 1.41·12-s + 0.679·13-s + (0.921 + 0.387i)14-s − 16-s + 1.18·17-s + (0.707 − 0.707i)18-s + (1.30 − 0.534i)21-s + (1.06 + 1.06i)22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.659 - 0.752i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.659 - 0.752i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.927438 + 0.420438i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.927438 + 0.420438i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1 - i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (1 + 2.44i)T \) |
| good | 3 | \( 1 - 2.44iT - 3T^{2} \) |
| 11 | \( 1 + 5iT - 11T^{2} \) |
| 13 | \( 1 - 2.44T + 13T^{2} \) |
| 17 | \( 1 - 4.89T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + T + 23T^{2} \) |
| 29 | \( 1 - 5T + 29T^{2} \) |
| 31 | \( 1 - 7.34T + 31T^{2} \) |
| 37 | \( 1 + 3iT - 37T^{2} \) |
| 41 | \( 1 + 12.2iT - 41T^{2} \) |
| 43 | \( 1 + 11T + 43T^{2} \) |
| 47 | \( 1 + 4.89iT - 47T^{2} \) |
| 53 | \( 1 - 4iT - 53T^{2} \) |
| 59 | \( 1 - 12.2T + 59T^{2} \) |
| 61 | \( 1 - 12.2iT - 61T^{2} \) |
| 67 | \( 1 - 3T + 67T^{2} \) |
| 71 | \( 1 - 5iT - 71T^{2} \) |
| 73 | \( 1 - 2.44T + 73T^{2} \) |
| 79 | \( 1 + 9iT - 79T^{2} \) |
| 83 | \( 1 - 2.44iT - 83T^{2} \) |
| 89 | \( 1 + 2.44iT - 89T^{2} \) |
| 97 | \( 1 + 7.34T + 97T^{2} \) |
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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
−10.28697725390078241355762702281, −9.888644133298439386535891692737, −8.766222993086115063214528478878, −8.289387031985464566352725586483, −7.08684948045845682130025427804, −6.04746270182600918334889341751, −5.29208683329478640771215784511, −4.11424542553821051055063356186, −3.25674459130478828005607176175, −0.825728559347057657812503843887,
1.22102541568391849971256756268, 2.20112813303573228190374159614, 3.20676965909006776525020044636, 4.79964732481931439554506576631, 6.31628382333336540659434826476, 6.89070177980392820890008859170, 8.022408293692576351093432269579, 8.332734416431613747582750609054, 9.686388246471395816648402674712, 10.04358915628603468833874212187
