L(s) = 1 | + (0.587 − 0.809i)9-s + (−1.76 + 0.278i)13-s + (0.896 − 1.76i)17-s + (1.11 + 0.363i)29-s + (−0.0489 − 0.309i)37-s + (−1.53 − 1.11i)41-s − i·49-s + (−0.412 − 0.809i)53-s + (1.53 − 1.11i)61-s + (−0.142 + 0.896i)73-s + (−0.309 − 0.951i)81-s + (0.690 + 0.951i)89-s + (0.278 − 0.142i)97-s + 1.61·101-s + (−0.363 + 0.5i)109-s + ⋯ |
L(s) = 1 | + (0.587 − 0.809i)9-s + (−1.76 + 0.278i)13-s + (0.896 − 1.76i)17-s + (1.11 + 0.363i)29-s + (−0.0489 − 0.309i)37-s + (−1.53 − 1.11i)41-s − i·49-s + (−0.412 − 0.809i)53-s + (1.53 − 1.11i)61-s + (−0.142 + 0.896i)73-s + (−0.309 − 0.951i)81-s + (0.690 + 0.951i)89-s + (0.278 − 0.142i)97-s + 1.61·101-s + (−0.363 + 0.5i)109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.349 + 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.349 + 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.155501637\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.155501637\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + (-0.587 + 0.809i)T^{2} \) |
| 7 | \( 1 + iT^{2} \) |
| 11 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 13 | \( 1 + (1.76 - 0.278i)T + (0.951 - 0.309i)T^{2} \) |
| 17 | \( 1 + (-0.896 + 1.76i)T + (-0.587 - 0.809i)T^{2} \) |
| 19 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 23 | \( 1 + (-0.951 - 0.309i)T^{2} \) |
| 29 | \( 1 + (-1.11 - 0.363i)T + (0.809 + 0.587i)T^{2} \) |
| 31 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 37 | \( 1 + (0.0489 + 0.309i)T + (-0.951 + 0.309i)T^{2} \) |
| 41 | \( 1 + (1.53 + 1.11i)T + (0.309 + 0.951i)T^{2} \) |
| 43 | \( 1 - iT^{2} \) |
| 47 | \( 1 + (0.587 - 0.809i)T^{2} \) |
| 53 | \( 1 + (0.412 + 0.809i)T + (-0.587 + 0.809i)T^{2} \) |
| 59 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 61 | \( 1 + (-1.53 + 1.11i)T + (0.309 - 0.951i)T^{2} \) |
| 67 | \( 1 + (-0.587 - 0.809i)T^{2} \) |
| 71 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 73 | \( 1 + (0.142 - 0.896i)T + (-0.951 - 0.309i)T^{2} \) |
| 79 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 83 | \( 1 + (0.587 + 0.809i)T^{2} \) |
| 89 | \( 1 + (-0.690 - 0.951i)T + (-0.309 + 0.951i)T^{2} \) |
| 97 | \( 1 + (-0.278 + 0.142i)T + (0.587 - 0.809i)T^{2} \) |
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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
−8.540618750952022167753430638381, −7.54779036353424888098306991706, −7.05468925532623049434020511854, −6.52615789206189107973587363491, −5.14174828721281966561032318406, −5.01509339661478388007893203552, −3.83364520196653666878542299588, −3.00156338490065672218981955566, −2.07700804261814804163447586923, −0.66706664084150803662014498185,
1.40918208346695072936083268714, 2.36816929672583738871697376756, 3.29099709857617645181734339917, 4.39794545880960487511978560322, 4.91637179239567123045069275797, 5.77885180374325689918825911387, 6.60615365631797653007112430295, 7.46655943476229902502618431901, 7.945961478198880127290574118842, 8.594495559515315754026730682364