L(s) = 1 | + (−0.707 + 0.707i)2-s − 1.00i·4-s + (−1.88 + 1.20i)5-s + (−1.61 − 1.61i)7-s + (0.707 + 0.707i)8-s + (0.483 − 2.18i)10-s + 3.53i·11-s + (−0.800 + 0.800i)13-s + 2.28·14-s − 1.00·16-s + (−3.53 + 3.53i)17-s + 2.63i·19-s + (1.20 + 1.88i)20-s + (−2.49 − 2.49i)22-s + (−0.707 − 0.707i)23-s + ⋯ |
L(s) = 1 | + (−0.499 + 0.499i)2-s − 0.500i·4-s + (−0.843 + 0.537i)5-s + (−0.609 − 0.609i)7-s + (0.250 + 0.250i)8-s + (0.152 − 0.690i)10-s + 1.06i·11-s + (−0.221 + 0.221i)13-s + 0.609·14-s − 0.250·16-s + (−0.857 + 0.857i)17-s + 0.603i·19-s + (0.268 + 0.421i)20-s + (−0.532 − 0.532i)22-s + (−0.147 − 0.147i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.164 + 0.986i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.164 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2461991307\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2461991307\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.707 - 0.707i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (1.88 - 1.20i)T \) |
| 23 | \( 1 + (0.707 + 0.707i)T \) |
good | 7 | \( 1 + (1.61 + 1.61i)T + 7iT^{2} \) |
| 11 | \( 1 - 3.53iT - 11T^{2} \) |
| 13 | \( 1 + (0.800 - 0.800i)T - 13iT^{2} \) |
| 17 | \( 1 + (3.53 - 3.53i)T - 17iT^{2} \) |
| 19 | \( 1 - 2.63iT - 19T^{2} \) |
| 29 | \( 1 - 6.66T + 29T^{2} \) |
| 31 | \( 1 - 4.22T + 31T^{2} \) |
| 37 | \( 1 + (4.02 + 4.02i)T + 37iT^{2} \) |
| 41 | \( 1 - 0.501iT - 41T^{2} \) |
| 43 | \( 1 + (4.11 - 4.11i)T - 43iT^{2} \) |
| 47 | \( 1 + (-3.72 + 3.72i)T - 47iT^{2} \) |
| 53 | \( 1 + (-2.30 - 2.30i)T + 53iT^{2} \) |
| 59 | \( 1 + 12.5T + 59T^{2} \) |
| 61 | \( 1 + 13.2T + 61T^{2} \) |
| 67 | \( 1 + (5.47 + 5.47i)T + 67iT^{2} \) |
| 71 | \( 1 + 10.0iT - 71T^{2} \) |
| 73 | \( 1 + (-9.57 + 9.57i)T - 73iT^{2} \) |
| 79 | \( 1 + 0.108iT - 79T^{2} \) |
| 83 | \( 1 + (5.67 + 5.67i)T + 83iT^{2} \) |
| 89 | \( 1 - 3.06T + 89T^{2} \) |
| 97 | \( 1 + (7.17 + 7.17i)T + 97iT^{2} \) |
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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
−8.860473035858321489551745357347, −8.012472102006662005553621129975, −7.41837207537759061142347201225, −6.65899803019056861409944714631, −6.23043735602104316306224710517, −4.73447042517511397778971313819, −4.16659111630442789311588106978, −3.09536096175188035611797961524, −1.80691661277028107596804475486, −0.12734739065904289125883995199,
0.974021941116558616487650723813, 2.63982005539356511024780112510, 3.21724790614863949428219708697, 4.34676597919518496740980534862, 5.14771037894840954891933462547, 6.26109411916743768507338006701, 7.06852450695678252401725964954, 8.000245031173796931954114841015, 8.706175225936613711967008132335, 9.081700146126117661297731760185