L(s) = 1 | + (0.445 + 0.895i)2-s + (−0.602 + 0.798i)4-s + (1.93 − 0.361i)5-s + (−0.982 − 0.183i)8-s + (0.739 + 0.673i)9-s + (1.18 + 1.56i)10-s + (−1.45 − 0.271i)13-s + (−0.273 − 0.961i)16-s + (−0.273 − 0.961i)17-s + (−0.273 + 0.961i)18-s + (−0.876 + 1.75i)20-s + (2.67 − 1.03i)25-s + (−0.404 − 1.42i)26-s + (−0.890 + 1.17i)29-s + (0.739 − 0.673i)32-s + ⋯ |
L(s) = 1 | + (0.445 + 0.895i)2-s + (−0.602 + 0.798i)4-s + (1.93 − 0.361i)5-s + (−0.982 − 0.183i)8-s + (0.739 + 0.673i)9-s + (1.18 + 1.56i)10-s + (−1.45 − 0.271i)13-s + (−0.273 − 0.961i)16-s + (−0.273 − 0.961i)17-s + (−0.273 + 0.961i)18-s + (−0.876 + 1.75i)20-s + (2.67 − 1.03i)25-s + (−0.404 − 1.42i)26-s + (−0.890 + 1.17i)29-s + (0.739 − 0.673i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.304 - 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.304 - 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.600242493\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.600242493\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.445 - 0.895i)T \) |
| 17 | \( 1 + (0.273 + 0.961i)T \) |
good | 3 | \( 1 + (-0.739 - 0.673i)T^{2} \) |
| 5 | \( 1 + (-1.93 + 0.361i)T + (0.932 - 0.361i)T^{2} \) |
| 7 | \( 1 + (-0.0922 - 0.995i)T^{2} \) |
| 11 | \( 1 + (0.273 + 0.961i)T^{2} \) |
| 13 | \( 1 + (1.45 + 0.271i)T + (0.932 + 0.361i)T^{2} \) |
| 19 | \( 1 + (0.602 + 0.798i)T^{2} \) |
| 23 | \( 1 + (-0.0922 - 0.995i)T^{2} \) |
| 29 | \( 1 + (0.890 - 1.17i)T + (-0.273 - 0.961i)T^{2} \) |
| 31 | \( 1 + (-0.932 - 0.361i)T^{2} \) |
| 37 | \( 1 + (0.757 + 0.469i)T + (0.445 + 0.895i)T^{2} \) |
| 41 | \( 1 + (1.83 + 0.710i)T + (0.739 + 0.673i)T^{2} \) |
| 43 | \( 1 + (0.850 + 0.526i)T^{2} \) |
| 47 | \( 1 + (-0.0922 + 0.995i)T^{2} \) |
| 53 | \( 1 + (-0.136 - 0.124i)T + (0.0922 + 0.995i)T^{2} \) |
| 59 | \( 1 + (0.982 + 0.183i)T^{2} \) |
| 61 | \( 1 + (-0.172 - 1.85i)T + (-0.982 + 0.183i)T^{2} \) |
| 67 | \( 1 + (0.602 + 0.798i)T^{2} \) |
| 71 | \( 1 + (-0.0922 - 0.995i)T^{2} \) |
| 73 | \( 1 + (0.404 + 1.42i)T + (-0.850 + 0.526i)T^{2} \) |
| 79 | \( 1 + (0.602 + 0.798i)T^{2} \) |
| 83 | \( 1 + (-0.739 + 0.673i)T^{2} \) |
| 89 | \( 1 + (0.876 - 0.163i)T + (0.932 - 0.361i)T^{2} \) |
| 97 | \( 1 + (0.890 + 0.811i)T + (0.0922 + 0.995i)T^{2} \) |
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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
−9.916589352822742420383789659449, −9.321553051667462363877920548427, −8.604281509717314028999239275185, −7.26924744925333315087420438910, −6.97120557832501229053591965582, −5.73789840424336808113184440999, −5.17889292700883542755905128143, −4.60317115445292823801033797380, −2.90764237320056273986029850714, −1.88886315088026377400314852111,
1.65715191141002781406647668283, 2.28110510539716804046810084134, 3.46564918610721445163387014426, 4.68552091801567158919153803852, 5.47197271719074349378145596950, 6.35249290787442743983206304397, 6.95403178762972784756967012829, 8.567917414102033230034570677089, 9.574992826529639529705125653143, 9.851406553907865463510111650600