L(s) = 1 | + i·2-s + 3.23i·3-s − 4-s − 3.23·6-s + i·7-s − i·8-s − 7.47·9-s + 11-s − 3.23i·12-s − 1.23i·13-s − 14-s + 16-s − 6.47i·17-s − 7.47i·18-s + 2.76·19-s + ⋯ |
L(s) = 1 | + 0.707i·2-s + 1.86i·3-s − 0.5·4-s − 1.32·6-s + 0.377i·7-s − 0.353i·8-s − 2.49·9-s + 0.301·11-s − 0.934i·12-s − 0.342i·13-s − 0.267·14-s + 0.250·16-s − 1.56i·17-s − 1.76i·18-s + 0.634·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.288420580\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.288420580\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 - iT \) |
| 11 | \( 1 - T \) |
good | 3 | \( 1 - 3.23iT - 3T^{2} \) |
| 13 | \( 1 + 1.23iT - 13T^{2} \) |
| 17 | \( 1 + 6.47iT - 17T^{2} \) |
| 19 | \( 1 - 2.76T + 19T^{2} \) |
| 23 | \( 1 + 4iT - 23T^{2} \) |
| 29 | \( 1 - 4.47T + 29T^{2} \) |
| 31 | \( 1 - 2T + 31T^{2} \) |
| 37 | \( 1 + 10.9iT - 37T^{2} \) |
| 41 | \( 1 - 6.47T + 41T^{2} \) |
| 43 | \( 1 - 1.52iT - 43T^{2} \) |
| 47 | \( 1 + 2iT - 47T^{2} \) |
| 53 | \( 1 - 0.472iT - 53T^{2} \) |
| 59 | \( 1 + 7.23T + 59T^{2} \) |
| 61 | \( 1 + 5.23T + 61T^{2} \) |
| 67 | \( 1 + 15.4iT - 67T^{2} \) |
| 71 | \( 1 + 2.47T + 71T^{2} \) |
| 73 | \( 1 - 4.94iT - 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 10.1iT - 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 - 3.52iT - 97T^{2} \) |
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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
−8.810496139430503828366458435134, −8.044392298461557007322928285487, −7.16680783306737814128387567258, −6.12107857098069271303632033745, −5.52189541179605798737979062990, −4.79218568564416487079530247264, −4.32906079563726233520875573296, −3.30437871273578586488741997646, −2.64773738519335148180714783071, −0.42917313236932262962014904527,
1.11449608894753470508524368225, 1.53257578986277262814719196978, 2.59301432167620350666415976450, 3.40320638655436539252217643596, 4.43645495969863769291326520343, 5.59050799345235651764144532466, 6.26559574104444710459687645517, 6.89375078520049401602844358511, 7.71895336177814404590453646172, 8.227359210602016317324017755805