| L(s) = 1 | + i·2-s − 4-s + i·7-s − i·8-s + 5·11-s − i·13-s − 14-s + 16-s + 5i·17-s + 5i·22-s + 26-s − i·28-s − 7·29-s − 9·31-s + i·32-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.5·4-s + 0.377i·7-s − 0.353i·8-s + 1.50·11-s − 0.277i·13-s − 0.267·14-s + 0.250·16-s + 1.21i·17-s + 1.06i·22-s + 0.196·26-s − 0.188i·28-s − 1.29·29-s − 1.61·31-s + 0.176i·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.432850666\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.432850666\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 + iT \) |
| good | 7 | \( 1 - iT - 7T^{2} \) |
| 11 | \( 1 - 5T + 11T^{2} \) |
| 17 | \( 1 - 5iT - 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 + 7T + 29T^{2} \) |
| 31 | \( 1 + 9T + 31T^{2} \) |
| 37 | \( 1 - 8iT - 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 - 8iT - 43T^{2} \) |
| 47 | \( 1 + 9iT - 47T^{2} \) |
| 53 | \( 1 + 11iT - 53T^{2} \) |
| 59 | \( 1 - T + 59T^{2} \) |
| 61 | \( 1 + 7T + 61T^{2} \) |
| 67 | \( 1 - 15iT - 67T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 - 4iT - 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 - 9iT - 83T^{2} \) |
| 89 | \( 1 - 16T + 89T^{2} \) |
| 97 | \( 1 + 2iT - 97T^{2} \) |
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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
−8.434728547463988984308788074081, −7.65860223445967591954107841383, −6.90747533466767922952746959945, −6.29189856495023502658141869488, −5.69436074041365981768891564556, −4.94448647757579106848986553461, −3.87112615257391109495795722558, −3.58660431432969335017274393453, −2.12695607920601683592955973037, −1.19686538178669012952217751678,
0.39419801339955705971946892769, 1.47968324045681644356664269715, 2.29266601285996663458552866241, 3.45960318694380298751265032289, 3.92604037875852531781282651217, 4.70775712391222487495970690070, 5.59454489145823828092195379654, 6.34210447396400464673866349846, 7.34324858907728589166864529229, 7.55292503815709720826040571861
