L(s) = 1 | + i·2-s + 1.53i·3-s − 4-s − 1.53·6-s + 2.87i·7-s − i·8-s + 0.630·9-s − 1.09·11-s − 1.53i·12-s + 4.53i·13-s − 2.87·14-s + 16-s − 2.80i·17-s + 0.630i·18-s + 5.04·19-s + ⋯ |
L(s) = 1 | + 0.707i·2-s + 0.888i·3-s − 0.5·4-s − 0.628·6-s + 1.08i·7-s − 0.353i·8-s + 0.210·9-s − 0.329·11-s − 0.444i·12-s + 1.25i·13-s − 0.769·14-s + 0.250·16-s − 0.679i·17-s + 0.148i·18-s + 1.15·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.367934458\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.367934458\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 5 | \( 1 \) |
| 37 | \( 1 + iT \) |
good | 3 | \( 1 - 1.53iT - 3T^{2} \) |
| 7 | \( 1 - 2.87iT - 7T^{2} \) |
| 11 | \( 1 + 1.09T + 11T^{2} \) |
| 13 | \( 1 - 4.53iT - 13T^{2} \) |
| 17 | \( 1 + 2.80iT - 17T^{2} \) |
| 19 | \( 1 - 5.04T + 19T^{2} \) |
| 23 | \( 1 - 7.41iT - 23T^{2} \) |
| 29 | \( 1 + 6.68T + 29T^{2} \) |
| 31 | \( 1 - 3.51T + 31T^{2} \) |
| 41 | \( 1 + 8.07T + 41T^{2} \) |
| 43 | \( 1 - 10.2iT - 43T^{2} \) |
| 47 | \( 1 + 8.68iT - 47T^{2} \) |
| 53 | \( 1 + 10.0iT - 53T^{2} \) |
| 59 | \( 1 + 10.2T + 59T^{2} \) |
| 61 | \( 1 - 6.29T + 61T^{2} \) |
| 67 | \( 1 + 13.2iT - 67T^{2} \) |
| 71 | \( 1 - 6.29T + 71T^{2} \) |
| 73 | \( 1 - 12.7iT - 73T^{2} \) |
| 79 | \( 1 + 2.58T + 79T^{2} \) |
| 83 | \( 1 - 8.48iT - 83T^{2} \) |
| 89 | \( 1 - 6.51T + 89T^{2} \) |
| 97 | \( 1 + 3.07iT - 97T^{2} \) |
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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
−9.439478341381714955986585437475, −9.189117603180988001893832540409, −8.141975143802460841840960808908, −7.30059070171358478582268189049, −6.54603635400240856125697308244, −5.33764509727001386294278482465, −5.16087911403662065132280884274, −4.02925204425138839554718935885, −3.16547391109303305961614879876, −1.74387961788892962886322274658,
0.53370545389602944928403882650, 1.42400108304827370403077548257, 2.63219155795501243909286484836, 3.61322112380738309964549347529, 4.51829684238159784141421063304, 5.53358752039962179780261788113, 6.50522988321147827929697847086, 7.45037522205257187897049266835, 7.83755960583173033190996020582, 8.719205433711584217520713483751