| L(s) = 1 | + (−0.691 − 0.866i)2-s + (0.988 + 0.149i)3-s + (0.171 − 0.751i)4-s + (2.06 + 1.41i)5-s + (−0.554 − 0.959i)6-s + (0.151 − 0.262i)7-s + (−2.76 + 1.33i)8-s + (0.955 + 0.294i)9-s + (−0.207 − 2.76i)10-s + (−0.0162 − 0.0710i)11-s + (0.281 − 0.717i)12-s + (0.336 − 4.48i)13-s + (−0.332 + 0.0500i)14-s + (1.83 + 1.70i)15-s + (1.67 + 0.808i)16-s + (−2.38 + 1.62i)17-s + ⋯ |
| L(s) = 1 | + (−0.488 − 0.612i)2-s + (0.570 + 0.0860i)3-s + (0.0858 − 0.375i)4-s + (0.925 + 0.630i)5-s + (−0.226 − 0.391i)6-s + (0.0572 − 0.0991i)7-s + (−0.978 + 0.471i)8-s + (0.318 + 0.0982i)9-s + (−0.0655 − 0.875i)10-s + (−0.00488 − 0.0214i)11-s + (0.0813 − 0.207i)12-s + (0.0932 − 1.24i)13-s + (−0.0887 + 0.0133i)14-s + (0.473 + 0.439i)15-s + (0.419 + 0.202i)16-s + (−0.578 + 0.394i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 129 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.661 + 0.749i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 129 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.661 + 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.01729 - 0.458966i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.01729 - 0.458966i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.988 - 0.149i)T \) |
| 43 | \( 1 + (5.24 - 3.92i)T \) |
| good | 2 | \( 1 + (0.691 + 0.866i)T + (-0.445 + 1.94i)T^{2} \) |
| 5 | \( 1 + (-2.06 - 1.41i)T + (1.82 + 4.65i)T^{2} \) |
| 7 | \( 1 + (-0.151 + 0.262i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (0.0162 + 0.0710i)T + (-9.91 + 4.77i)T^{2} \) |
| 13 | \( 1 + (-0.336 + 4.48i)T + (-12.8 - 1.93i)T^{2} \) |
| 17 | \( 1 + (2.38 - 1.62i)T + (6.21 - 15.8i)T^{2} \) |
| 19 | \( 1 + (-0.233 + 0.0718i)T + (15.6 - 10.7i)T^{2} \) |
| 23 | \( 1 + (5.44 - 5.04i)T + (1.71 - 22.9i)T^{2} \) |
| 29 | \( 1 + (-2.99 + 0.451i)T + (27.7 - 8.54i)T^{2} \) |
| 31 | \( 1 + (3.60 - 9.18i)T + (-22.7 - 21.0i)T^{2} \) |
| 37 | \( 1 + (-1.18 - 2.05i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (2.16 + 2.72i)T + (-9.12 + 39.9i)T^{2} \) |
| 47 | \( 1 + (-2.39 + 10.4i)T + (-42.3 - 20.3i)T^{2} \) |
| 53 | \( 1 + (0.179 + 2.39i)T + (-52.4 + 7.89i)T^{2} \) |
| 59 | \( 1 + (-0.192 - 0.0924i)T + (36.7 + 46.1i)T^{2} \) |
| 61 | \( 1 + (2.33 + 5.94i)T + (-44.7 + 41.4i)T^{2} \) |
| 67 | \( 1 + (-6.95 + 2.14i)T + (55.3 - 37.7i)T^{2} \) |
| 71 | \( 1 + (2.84 + 2.64i)T + (5.30 + 70.8i)T^{2} \) |
| 73 | \( 1 + (-0.512 + 6.83i)T + (-72.1 - 10.8i)T^{2} \) |
| 79 | \( 1 + (-5.59 + 9.69i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (14.7 + 2.22i)T + (79.3 + 24.4i)T^{2} \) |
| 89 | \( 1 + (-12.6 - 1.91i)T + (85.0 + 26.2i)T^{2} \) |
| 97 | \( 1 + (-3.95 - 17.3i)T + (-87.3 + 42.0i)T^{2} \) |
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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
−13.34221172474476521693038348050, −12.02454756424272989319423126418, −10.64078569614315776489653962432, −10.25127780929420051229647102625, −9.266182592209332385320585329245, −8.125466631943067881236325101090, −6.54832012434896141270173583396, −5.42427850570349522212850709720, −3.21367107583984218778630840255, −1.89832312018182987085758379384,
2.26548087171336642085263601779, 4.23052785837905140386740814051, 5.99653628350379422834586336595, 7.05056434923013581379415661972, 8.310476439469084841409959491912, 9.090001686644344504211332698025, 9.819381582679214212173196986153, 11.56094552280789968870404217546, 12.62365619266256171754415087539, 13.52669082344975800325948829498
