L(s) = 1 | + (−1.42 + 1.42i)3-s + (1.72 − 1.42i)5-s − 4.91i·7-s − 1.06i·9-s + (0.276 + 0.276i)11-s + (3.42 + 1.12i)13-s + (−0.424 + 4.48i)15-s + (3.14 − 3.14i)17-s + (2.48 + 2.48i)19-s + (6.99 + 6.99i)21-s + (−1.27 − 1.27i)23-s + (0.938 − 4.91i)25-s + (−2.76 − 2.76i)27-s + 5.46i·29-s + (1.12 − 1.12i)31-s + ⋯ |
L(s) = 1 | + (−0.822 + 0.822i)3-s + (0.770 − 0.637i)5-s − 1.85i·7-s − 0.353i·9-s + (0.0834 + 0.0834i)11-s + (0.949 + 0.312i)13-s + (−0.109 + 1.15i)15-s + (0.763 − 0.763i)17-s + (0.570 + 0.570i)19-s + (1.52 + 1.52i)21-s + (−0.266 − 0.266i)23-s + (0.187 − 0.982i)25-s + (−0.531 − 0.531i)27-s + 1.01i·29-s + (0.202 − 0.202i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.10676 - 0.195741i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.10676 - 0.195741i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-1.72 + 1.42i)T \) |
| 13 | \( 1 + (-3.42 - 1.12i)T \) |
good | 3 | \( 1 + (1.42 - 1.42i)T - 3iT^{2} \) |
| 7 | \( 1 + 4.91iT - 7T^{2} \) |
| 11 | \( 1 + (-0.276 - 0.276i)T + 11iT^{2} \) |
| 17 | \( 1 + (-3.14 + 3.14i)T - 17iT^{2} \) |
| 19 | \( 1 + (-2.48 - 2.48i)T + 19iT^{2} \) |
| 23 | \( 1 + (1.27 + 1.27i)T + 23iT^{2} \) |
| 29 | \( 1 - 5.46iT - 29T^{2} \) |
| 31 | \( 1 + (-1.12 + 1.12i)T - 31iT^{2} \) |
| 37 | \( 1 + 1.93iT - 37T^{2} \) |
| 41 | \( 1 + (0.237 - 0.237i)T - 41iT^{2} \) |
| 43 | \( 1 + (6.16 + 6.16i)T + 43iT^{2} \) |
| 47 | \( 1 - 13.3iT - 47T^{2} \) |
| 53 | \( 1 + (-2.50 + 2.50i)T - 53iT^{2} \) |
| 59 | \( 1 + (-2.48 + 2.48i)T - 59iT^{2} \) |
| 61 | \( 1 + 1.25T + 61T^{2} \) |
| 67 | \( 1 - 9.76T + 67T^{2} \) |
| 71 | \( 1 + (3.61 - 3.61i)T - 71iT^{2} \) |
| 73 | \( 1 + 12.6T + 73T^{2} \) |
| 79 | \( 1 - 8.07iT - 79T^{2} \) |
| 83 | \( 1 - 11.2iT - 83T^{2} \) |
| 89 | \( 1 + (-8.16 + 8.16i)T - 89iT^{2} \) |
| 97 | \( 1 + 16.3T + 97T^{2} \) |
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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
−11.72227724835633028658593825256, −10.76920739037190308197033745957, −10.16646980948147828087604742173, −9.439467501722652380224366828765, −8.024446166381580447574271337245, −6.82239853866242915105361193499, −5.65040418567341855585470419539, −4.69644136395765018055023067431, −3.73926103348645964961103367730, −1.14228121568982728795594409616,
1.72616182262416600700108333457, 3.12289038825013155740295364735, 5.48909518327577227843558348305, 5.91600732728809107442965416198, 6.72623895422650076075454590624, 8.142584131919569121500047844142, 9.168405995114640729375155949193, 10.20306736409358524936832903021, 11.45619459524647661034287513528, 11.88888371686150879481179947345