L(s) = 1 | + (−0.766 + 0.642i)4-s + (0.766 + 0.642i)5-s + (0.939 + 0.342i)9-s + (1 − 1.73i)11-s + (0.173 − 0.984i)16-s − 20-s + (0.173 + 0.984i)25-s + (−0.939 + 0.342i)36-s + (0.347 + 1.96i)44-s + (0.5 + 0.866i)45-s + (−0.5 + 0.866i)49-s + (1.87 − 0.684i)55-s + (−1.53 + 1.28i)61-s + (0.500 + 0.866i)64-s + (0.766 − 0.642i)80-s + (0.766 + 0.642i)81-s + ⋯ |
L(s) = 1 | + (−0.766 + 0.642i)4-s + (0.766 + 0.642i)5-s + (0.939 + 0.342i)9-s + (1 − 1.73i)11-s + (0.173 − 0.984i)16-s − 20-s + (0.173 + 0.984i)25-s + (−0.939 + 0.342i)36-s + (0.347 + 1.96i)44-s + (0.5 + 0.866i)45-s + (−0.5 + 0.866i)49-s + (1.87 − 0.684i)55-s + (−1.53 + 1.28i)61-s + (0.500 + 0.866i)64-s + (0.766 − 0.642i)80-s + (0.766 + 0.642i)81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.845 - 0.533i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.845 - 0.533i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.237812909\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.237812909\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (-0.766 - 0.642i)T \) |
| 19 | \( 1 \) |
good | 2 | \( 1 + (0.766 - 0.642i)T^{2} \) |
| 3 | \( 1 + (-0.939 - 0.342i)T^{2} \) |
| 7 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + (-0.939 + 0.342i)T^{2} \) |
| 17 | \( 1 + (-0.766 + 0.642i)T^{2} \) |
| 23 | \( 1 + (-0.173 + 0.984i)T^{2} \) |
| 29 | \( 1 + (-0.766 - 0.642i)T^{2} \) |
| 31 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( 1 + (0.939 + 0.342i)T^{2} \) |
| 43 | \( 1 + (-0.173 - 0.984i)T^{2} \) |
| 47 | \( 1 + (-0.766 - 0.642i)T^{2} \) |
| 53 | \( 1 + (0.173 - 0.984i)T^{2} \) |
| 59 | \( 1 + (-0.766 + 0.642i)T^{2} \) |
| 61 | \( 1 + (1.53 - 1.28i)T + (0.173 - 0.984i)T^{2} \) |
| 67 | \( 1 + (0.766 + 0.642i)T^{2} \) |
| 71 | \( 1 + (-0.173 - 0.984i)T^{2} \) |
| 73 | \( 1 + (0.939 + 0.342i)T^{2} \) |
| 79 | \( 1 + (0.939 + 0.342i)T^{2} \) |
| 83 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 89 | \( 1 + (0.939 - 0.342i)T^{2} \) |
| 97 | \( 1 + (0.766 - 0.642i)T^{2} \) |
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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
−9.394458300948746337032823232894, −8.880078676127905887333142455259, −8.012049050841537084367485005199, −7.17833978345114062448691787543, −6.33003101452018145984764282642, −5.57599873395924136688196209885, −4.48617816187546555322048469661, −3.62486903072526007638283161933, −2.84011568267798594093862487925, −1.34174102720537357181234126367,
1.27942351034480932263572465497, 1.97890772467771187159625120584, 3.85662630177217828697355848371, 4.58028323806090855647842254094, 5.11940476681394164538746860672, 6.25014219111365568491846767724, 6.82286687743419711480794502596, 7.87144543461986724597741938251, 9.009674578536898325808552518407, 9.402460319134091265747724107153