| L(s) = 1 | + (0.181 + 1.40i)2-s + (−0.214 + 0.459i)3-s + (−1.93 + 0.509i)4-s + (2.68 − 1.87i)5-s + (−0.683 − 0.217i)6-s + (0.233 + 0.404i)7-s + (−1.06 − 2.62i)8-s + (1.76 + 2.10i)9-s + (3.12 + 3.42i)10-s + (0.271 + 1.01i)11-s + (0.180 − 0.998i)12-s + (2.63 + 5.64i)13-s + (−0.524 + 0.400i)14-s + (0.288 + 1.63i)15-s + (3.48 − 1.97i)16-s + (0.0648 + 0.0544i)17-s + ⋯ |
| L(s) = 1 | + (0.128 + 0.991i)2-s + (−0.123 + 0.265i)3-s + (−0.967 + 0.254i)4-s + (1.19 − 0.839i)5-s + (−0.279 − 0.0886i)6-s + (0.0881 + 0.152i)7-s + (−0.376 − 0.926i)8-s + (0.587 + 0.700i)9-s + (0.986 + 1.08i)10-s + (0.0818 + 0.305i)11-s + (0.0520 − 0.288i)12-s + (0.729 + 1.56i)13-s + (−0.140 + 0.107i)14-s + (0.0744 + 0.422i)15-s + (0.870 − 0.492i)16-s + (0.0157 + 0.0132i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0111 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0111 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.06682 + 1.05500i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.06682 + 1.05500i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.181 - 1.40i)T \) |
| 19 | \( 1 + (4.15 + 1.32i)T \) |
| good | 3 | \( 1 + (0.214 - 0.459i)T + (-1.92 - 2.29i)T^{2} \) |
| 5 | \( 1 + (-2.68 + 1.87i)T + (1.71 - 4.69i)T^{2} \) |
| 7 | \( 1 + (-0.233 - 0.404i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.271 - 1.01i)T + (-9.52 + 5.5i)T^{2} \) |
| 13 | \( 1 + (-2.63 - 5.64i)T + (-8.35 + 9.95i)T^{2} \) |
| 17 | \( 1 + (-0.0648 - 0.0544i)T + (2.95 + 16.7i)T^{2} \) |
| 23 | \( 1 + (1.21 + 6.86i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (0.229 + 0.0201i)T + (28.5 + 5.03i)T^{2} \) |
| 31 | \( 1 + (-2.25 - 3.90i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-1.70 + 1.70i)T - 37iT^{2} \) |
| 41 | \( 1 + (1.18 - 0.429i)T + (31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (1.18 + 1.69i)T + (-14.7 + 40.4i)T^{2} \) |
| 47 | \( 1 + (-6.35 - 7.57i)T + (-8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 + (8.68 + 6.07i)T + (18.1 + 49.8i)T^{2} \) |
| 59 | \( 1 + (0.726 + 8.30i)T + (-58.1 + 10.2i)T^{2} \) |
| 61 | \( 1 + (7.16 + 5.01i)T + (20.8 + 57.3i)T^{2} \) |
| 67 | \( 1 + (-0.173 + 1.98i)T + (-65.9 - 11.6i)T^{2} \) |
| 71 | \( 1 + (-7.60 - 1.34i)T + (66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 + (4.64 + 12.7i)T + (-55.9 + 46.9i)T^{2} \) |
| 79 | \( 1 + (13.7 - 4.99i)T + (60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (-0.339 + 1.26i)T + (-71.8 - 41.5i)T^{2} \) |
| 89 | \( 1 + (-6.00 - 2.18i)T + (68.1 + 57.2i)T^{2} \) |
| 97 | \( 1 + (-1.50 + 1.78i)T + (-16.8 - 95.5i)T^{2} \) |
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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
−12.33277636355317271087340456200, −10.80170425120893317068291646668, −9.767303720707667389096589301341, −9.035759048088002766154791529578, −8.294347753591754464359784961245, −6.82414281349947571869364492972, −6.09552437766012925328388310764, −4.88566101505909060439696780213, −4.31285646458267540340207896636, −1.84779012197756675715406980297,
1.34570959454604081586394830883, 2.78959591094539227166369000552, 3.90844512253585912873061116660, 5.64598944010044861706593999749, 6.21631058674760775781628172619, 7.70649779497109733195761874221, 8.980331129489531441520190359869, 9.999264843789319978053247077767, 10.45882211697670317489371164841, 11.36691694529075387660200653063
