| L(s) = 1 | + i·2-s + 0.377·3-s − 4-s + 0.377i·6-s − 0.631·7-s − i·8-s − 2.85·9-s + 1.24·11-s − 0.377·12-s − 3.34i·13-s − 0.631i·14-s + 16-s + 3.10i·17-s − 2.85i·18-s + 5.97i·19-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s + 0.218·3-s − 0.5·4-s + 0.154i·6-s − 0.238·7-s − 0.353i·8-s − 0.952·9-s + 0.376·11-s − 0.109·12-s − 0.929i·13-s − 0.168i·14-s + 0.250·16-s + 0.753i·17-s − 0.673i·18-s + 1.37i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.674 + 0.737i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.674 + 0.737i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.05967957457\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.05967957457\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (4.48 + 4.10i)T \) |
| good | 3 | \( 1 - 0.377T + 3T^{2} \) |
| 7 | \( 1 + 0.631T + 7T^{2} \) |
| 11 | \( 1 - 1.24T + 11T^{2} \) |
| 13 | \( 1 + 3.34iT - 13T^{2} \) |
| 17 | \( 1 - 3.10iT - 17T^{2} \) |
| 19 | \( 1 - 5.97iT - 19T^{2} \) |
| 23 | \( 1 - 7.60iT - 23T^{2} \) |
| 29 | \( 1 + 9.57iT - 29T^{2} \) |
| 31 | \( 1 + 7.26iT - 31T^{2} \) |
| 41 | \( 1 + 8.45T + 41T^{2} \) |
| 43 | \( 1 - 4.86iT - 43T^{2} \) |
| 47 | \( 1 + 13.1T + 47T^{2} \) |
| 53 | \( 1 + 7.17T + 53T^{2} \) |
| 59 | \( 1 + 4.36iT - 59T^{2} \) |
| 61 | \( 1 - 2.14iT - 61T^{2} \) |
| 67 | \( 1 - 11.3T + 67T^{2} \) |
| 71 | \( 1 + 12.7T + 71T^{2} \) |
| 73 | \( 1 + 4.45T + 73T^{2} \) |
| 79 | \( 1 + 8.78iT - 79T^{2} \) |
| 83 | \( 1 + 6.63T + 83T^{2} \) |
| 89 | \( 1 + 13.7iT - 89T^{2} \) |
| 97 | \( 1 - 11.0iT - 97T^{2} \) |
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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
−9.732177257381854151354841747866, −8.781809480965672156856869297176, −7.934100041286610517741902793280, −7.74140563052525409991328586554, −6.28305832553885660688409128495, −5.96436455379851089674105456228, −5.10443260158575322528474793400, −3.83100685972183418335672811811, −3.24012881457679198462553131358, −1.75825513358752905353080452593,
0.02049597870610969938322169308, 1.60466491202286376960396342052, 2.80842342255032991523203622374, 3.36217892551620962077004042923, 4.68130142797322709975924136662, 5.15905503786223494039986552083, 6.61474287624563905652784673050, 6.91656447004958990703410260494, 8.441325087549957798251977838116, 8.754786121516848987780300814122
