L(s) = 1 | − i·2-s − 1.09i·3-s − 4-s + (1.74 + 1.40i)5-s − 1.09·6-s + 3.20i·7-s + i·8-s + 1.80·9-s + (1.40 − 1.74i)10-s + 3.82·11-s + 1.09i·12-s + 0.147i·13-s + 3.20·14-s + (1.53 − 1.90i)15-s + 16-s − 0.978i·17-s + ⋯ |
L(s) = 1 | − 0.707i·2-s − 0.631i·3-s − 0.5·4-s + (0.779 + 0.626i)5-s − 0.446·6-s + 1.21i·7-s + 0.353i·8-s + 0.600·9-s + (0.443 − 0.551i)10-s + 1.15·11-s + 0.315i·12-s + 0.0408i·13-s + 0.857·14-s + (0.395 − 0.492i)15-s + 0.250·16-s − 0.237i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.779 + 0.626i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.779 + 0.626i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.47317 - 0.518788i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.47317 - 0.518788i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + iT \) |
| 5 | \( 1 + (-1.74 - 1.40i)T \) |
| 37 | \( 1 + iT \) |
good | 3 | \( 1 + 1.09iT - 3T^{2} \) |
| 7 | \( 1 - 3.20iT - 7T^{2} \) |
| 11 | \( 1 - 3.82T + 11T^{2} \) |
| 13 | \( 1 - 0.147iT - 13T^{2} \) |
| 17 | \( 1 + 0.978iT - 17T^{2} \) |
| 19 | \( 1 + 2.67T + 19T^{2} \) |
| 23 | \( 1 + 2.33iT - 23T^{2} \) |
| 29 | \( 1 + 6.30T + 29T^{2} \) |
| 31 | \( 1 + 3.62T + 31T^{2} \) |
| 41 | \( 1 - 11.8T + 41T^{2} \) |
| 43 | \( 1 + 4.53iT - 43T^{2} \) |
| 47 | \( 1 + 6.23iT - 47T^{2} \) |
| 53 | \( 1 - 11.2iT - 53T^{2} \) |
| 59 | \( 1 + 6.92T + 59T^{2} \) |
| 61 | \( 1 - 10.4T + 61T^{2} \) |
| 67 | \( 1 + 2.80iT - 67T^{2} \) |
| 71 | \( 1 + 12.3T + 71T^{2} \) |
| 73 | \( 1 + 13.9iT - 73T^{2} \) |
| 79 | \( 1 + 15.6T + 79T^{2} \) |
| 83 | \( 1 - 13.5iT - 83T^{2} \) |
| 89 | \( 1 - 6.46T + 89T^{2} \) |
| 97 | \( 1 - 3.07iT - 97T^{2} \) |
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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
−11.38396474706542010707507023066, −10.46315506878631924668699683357, −9.380913206204474609254931622402, −8.917218935955195804693430905891, −7.45109674618638508649794750417, −6.43066531768240706951453334380, −5.60436303314580054316646592434, −4.06435275898140240826563018683, −2.56422612539408917833572604751, −1.67255219193717623295780646184,
1.36728266841516739548101221167, 3.88591168176641197008988604593, 4.45768228585491245306933874824, 5.69685582524099154213524804501, 6.72845607588232631973235919169, 7.63009365869304704992014017588, 8.925088436157635897131214446175, 9.572917605042809569052016822135, 10.29655996581753657860199428609, 11.29666810398148934615118130455