| L(s) = 1 | + i·2-s + 3.34i·3-s − 4-s − 3.34·6-s − 2.59i·7-s − i·8-s − 8.19·9-s + 4.74·11-s − 3.34i·12-s − 6.69i·13-s + 2.59·14-s + 16-s − 0.748i·17-s − 8.19i·18-s + 3.34·19-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s + 1.93i·3-s − 0.5·4-s − 1.36·6-s − 0.981i·7-s − 0.353i·8-s − 2.73·9-s + 1.43·11-s − 0.965i·12-s − 1.85i·13-s + 0.694·14-s + 0.250·16-s − 0.181i·17-s − 1.93i·18-s + 0.767·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.493063487\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.493063487\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 37 | \( 1 + iT \) |
| good | 3 | \( 1 - 3.34iT - 3T^{2} \) |
| 7 | \( 1 + 2.59iT - 7T^{2} \) |
| 11 | \( 1 - 4.74T + 11T^{2} \) |
| 13 | \( 1 + 6.69iT - 13T^{2} \) |
| 17 | \( 1 + 0.748iT - 17T^{2} \) |
| 19 | \( 1 - 3.34T + 19T^{2} \) |
| 23 | \( 1 + 1.49iT - 23T^{2} \) |
| 29 | \( 1 + 3.94T + 29T^{2} \) |
| 31 | \( 1 - 7.79T + 31T^{2} \) |
| 41 | \( 1 + 6.44T + 41T^{2} \) |
| 43 | \( 1 + 1.94iT - 43T^{2} \) |
| 47 | \( 1 - 1.84iT - 47T^{2} \) |
| 53 | \( 1 + 10.4iT - 53T^{2} \) |
| 59 | \( 1 + 5.84T + 59T^{2} \) |
| 61 | \( 1 - 7.94T + 61T^{2} \) |
| 67 | \( 1 - 1.84iT - 67T^{2} \) |
| 71 | \( 1 + 3.88T + 71T^{2} \) |
| 73 | \( 1 - 7.49iT - 73T^{2} \) |
| 79 | \( 1 - 16.5T + 79T^{2} \) |
| 83 | \( 1 + 15.2iT - 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 + 10.4iT - 97T^{2} \) |
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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
−9.547224448363969298173710539116, −8.626824557963557544338016490205, −8.032148600391575467830185139138, −6.96505204834021500628011980233, −5.96039721595275308192581933611, −5.24765048083822574140661566258, −4.47668719851879990752589853102, −3.71267573026869637458518926580, −3.12547332349056135810887699352, −0.63390855109511972533099685245,
1.21967624179701784548455434461, 1.84800448447057828332884720395, 2.71823127919882464886203233547, 3.86692569781038100380642785662, 5.17570632084132790491013927666, 6.23219735339953370725548089926, 6.60729877887078619460438436599, 7.49797461339906262648856106222, 8.439522367075178374916625913177, 9.054856411104924971369612347189
