L(s) = 1 | + (−0.786 + 0.618i)3-s + (−0.841 + 0.540i)4-s + (−0.981 − 0.189i)7-s + (0.235 − 0.971i)9-s + (0.327 − 0.945i)12-s + (0.627 − 1.81i)13-s + (0.415 − 0.909i)16-s + (−0.209 + 0.713i)19-s + (0.888 − 0.458i)21-s + (−0.327 + 0.945i)25-s + (0.415 + 0.909i)27-s + (0.928 − 0.371i)28-s + (0.544 − 0.627i)31-s + (0.327 + 0.945i)36-s + 1.96·37-s + ⋯ |
L(s) = 1 | + (−0.786 + 0.618i)3-s + (−0.841 + 0.540i)4-s + (−0.981 − 0.189i)7-s + (0.235 − 0.971i)9-s + (0.327 − 0.945i)12-s + (0.627 − 1.81i)13-s + (0.415 − 0.909i)16-s + (−0.209 + 0.713i)19-s + (0.888 − 0.458i)21-s + (−0.327 + 0.945i)25-s + (0.415 + 0.909i)27-s + (0.928 − 0.371i)28-s + (0.544 − 0.627i)31-s + (0.327 + 0.945i)36-s + 1.96·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1407 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.954 + 0.297i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1407 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.954 + 0.297i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5394843065\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5394843065\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.786 - 0.618i)T \) |
| 7 | \( 1 + (0.981 + 0.189i)T \) |
| 67 | \( 1 + (0.981 - 0.189i)T \) |
good | 2 | \( 1 + (0.841 - 0.540i)T^{2} \) |
| 5 | \( 1 + (0.327 - 0.945i)T^{2} \) |
| 11 | \( 1 + (-0.654 - 0.755i)T^{2} \) |
| 13 | \( 1 + (-0.627 + 1.81i)T + (-0.786 - 0.618i)T^{2} \) |
| 17 | \( 1 + (0.580 - 0.814i)T^{2} \) |
| 19 | \( 1 + (0.209 - 0.713i)T + (-0.841 - 0.540i)T^{2} \) |
| 23 | \( 1 + (0.959 - 0.281i)T^{2} \) |
| 29 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 + (-0.544 + 0.627i)T + (-0.142 - 0.989i)T^{2} \) |
| 37 | \( 1 - 1.96T + T^{2} \) |
| 41 | \( 1 + (-0.580 + 0.814i)T^{2} \) |
| 43 | \( 1 + (-0.817 + 1.27i)T + (-0.415 - 0.909i)T^{2} \) |
| 47 | \( 1 + (0.235 - 0.971i)T^{2} \) |
| 53 | \( 1 + (0.580 + 0.814i)T^{2} \) |
| 59 | \( 1 + (0.928 + 0.371i)T^{2} \) |
| 61 | \( 1 + (0.771 + 1.68i)T + (-0.654 + 0.755i)T^{2} \) |
| 71 | \( 1 + (0.995 - 0.0950i)T^{2} \) |
| 73 | \( 1 + (-0.154 + 1.62i)T + (-0.981 - 0.189i)T^{2} \) |
| 79 | \( 1 + (-0.357 - 0.123i)T + (0.786 + 0.618i)T^{2} \) |
| 83 | \( 1 + (0.981 - 0.189i)T^{2} \) |
| 89 | \( 1 + (0.235 + 0.971i)T^{2} \) |
| 97 | \( 1 + (0.786 + 1.36i)T + (-0.5 + 0.866i)T^{2} \) |
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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
−9.753307504869741309727882461104, −9.121650881942513946089588063391, −8.123229017607925610638215242756, −7.39043046538568201890217681819, −6.07223032664008355394529099887, −5.69127137358146195694214582592, −4.58346559615588299095831038465, −3.68840967254920117682529245941, −3.10190462943463848638649673869, −0.62502996856896733153832024092,
1.11932982030622035772854814345, 2.50072217342111410854292656830, 4.13929291666599991459764926166, 4.68107181174604205664985453749, 5.98918315549168453831277301586, 6.27432360936979230219687036036, 7.11718058543501518921126304710, 8.291116724575876832325838340600, 9.122243733326807434219495793451, 9.700750198381854920024223769582