L(s) = 1 | + (0.707 + 0.707i)3-s + (0.707 − 0.707i)5-s + (1 + i)7-s + 1.00i·9-s − 1.41·11-s + 1.00·15-s + 1.41i·21-s − 1.00i·25-s + (−0.707 + 0.707i)27-s + 1.41·29-s + 2i·31-s + (−1.00 − 1.00i)33-s + 1.41·35-s + (0.707 + 0.707i)45-s + i·49-s + ⋯ |
L(s) = 1 | + (0.707 + 0.707i)3-s + (0.707 − 0.707i)5-s + (1 + i)7-s + 1.00i·9-s − 1.41·11-s + 1.00·15-s + 1.41i·21-s − 1.00i·25-s + (−0.707 + 0.707i)27-s + 1.41·29-s + 2i·31-s + (−1.00 − 1.00i)33-s + 1.41·35-s + (0.707 + 0.707i)45-s + i·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.525 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.525 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.948997776\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.948997776\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.707 - 0.707i)T \) |
| 5 | \( 1 + (-0.707 + 0.707i)T \) |
good | 7 | \( 1 + (-1 - i)T + iT^{2} \) |
| 11 | \( 1 + 1.41T + T^{2} \) |
| 13 | \( 1 + iT^{2} \) |
| 17 | \( 1 + iT^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 + iT^{2} \) |
| 29 | \( 1 - 1.41T + T^{2} \) |
| 31 | \( 1 - 2iT - T^{2} \) |
| 37 | \( 1 - iT^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - iT^{2} \) |
| 47 | \( 1 - iT^{2} \) |
| 53 | \( 1 - iT^{2} \) |
| 59 | \( 1 + 1.41iT - T^{2} \) |
| 61 | \( 1 + T^{2} \) |
| 67 | \( 1 + iT^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + (1 + i)T + iT^{2} \) |
| 79 | \( 1 - 2T + T^{2} \) |
| 83 | \( 1 + iT^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 + (1 - i)T - iT^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.695094140823356281507978953581, −8.294580541737325180440345131246, −7.70379917096916282235228414167, −6.46300073711219102476310578045, −5.33203675764183613089557409376, −5.13707773190869375960957411429, −4.49471012688057781765588230430, −3.12175028637615869916022753770, −2.42301765725169475677872051499, −1.61818054746038334790414496071,
1.07941476955759631796116144024, 2.21415576238961232477022316623, 2.73877668543896770791664711991, 3.82359985391496918586201932688, 4.74571584701478311387268655790, 5.68979692848576789645459137652, 6.46736101881538332885731275798, 7.29887102686644664630190000651, 7.74889659530919980626524336611, 8.277770774769356511051849188569