| L(s) = 1 | + (−4.89 − 0.256i)2-s + (5.67 + 7.00i)3-s + (15.9 + 1.67i)4-s + (−11.0 + 1.57i)5-s + (−25.9 − 35.7i)6-s + (17.8 − 5.00i)7-s + (−38.9 − 6.16i)8-s + (−11.2 + 52.9i)9-s + (54.6 − 4.85i)10-s + (46.2 − 9.84i)11-s + (78.7 + 121. i)12-s + (77.7 + 39.5i)13-s + (−88.6 + 19.9i)14-s + (−73.7 − 68.5i)15-s + (63.6 + 13.5i)16-s + (−1.45 − 3.79i)17-s + ⋯ |
| L(s) = 1 | + (−1.73 − 0.0907i)2-s + (1.09 + 1.34i)3-s + (1.99 + 0.209i)4-s + (−0.990 + 0.140i)5-s + (−1.76 − 2.43i)6-s + (0.962 − 0.270i)7-s + (−1.72 − 0.272i)8-s + (−0.417 + 1.96i)9-s + (1.72 − 0.153i)10-s + (1.26 − 0.269i)11-s + (1.89 + 2.91i)12-s + (1.65 + 0.844i)13-s + (−1.69 + 0.380i)14-s + (−1.26 − 1.18i)15-s + (0.994 + 0.211i)16-s + (−0.0207 − 0.0541i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.155 - 0.987i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.155 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.729415 + 0.853541i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.729415 + 0.853541i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (11.0 - 1.57i)T \) |
| 7 | \( 1 + (-17.8 + 5.00i)T \) |
| good | 2 | \( 1 + (4.89 + 0.256i)T + (7.95 + 0.836i)T^{2} \) |
| 3 | \( 1 + (-5.67 - 7.00i)T + (-5.61 + 26.4i)T^{2} \) |
| 11 | \( 1 + (-46.2 + 9.84i)T + (1.21e3 - 541. i)T^{2} \) |
| 13 | \( 1 + (-77.7 - 39.5i)T + (1.29e3 + 1.77e3i)T^{2} \) |
| 17 | \( 1 + (1.45 + 3.79i)T + (-3.65e3 + 3.28e3i)T^{2} \) |
| 19 | \( 1 + (-3.63 - 34.5i)T + (-6.70e3 + 1.42e3i)T^{2} \) |
| 23 | \( 1 + (-7.51 + 143. i)T + (-1.21e4 - 1.27e3i)T^{2} \) |
| 29 | \( 1 + (107. - 147. i)T + (-7.53e3 - 2.31e4i)T^{2} \) |
| 31 | \( 1 + (58.5 - 131. i)T + (-1.99e4 - 2.21e4i)T^{2} \) |
| 37 | \( 1 + (72.6 - 47.1i)T + (2.06e4 - 4.62e4i)T^{2} \) |
| 41 | \( 1 + (30.3 - 9.86i)T + (5.57e4 - 4.05e4i)T^{2} \) |
| 43 | \( 1 + (68.8 + 68.8i)T + 7.95e4iT^{2} \) |
| 47 | \( 1 + (198. + 76.3i)T + (7.71e4 + 6.94e4i)T^{2} \) |
| 53 | \( 1 + (92.6 - 75.0i)T + (3.09e4 - 1.45e5i)T^{2} \) |
| 59 | \( 1 + (403. - 448. i)T + (-2.14e4 - 2.04e5i)T^{2} \) |
| 61 | \( 1 + (315. - 284. i)T + (2.37e4 - 2.25e5i)T^{2} \) |
| 67 | \( 1 + (-301. + 115. i)T + (2.23e5 - 2.01e5i)T^{2} \) |
| 71 | \( 1 + (-471. - 342. i)T + (1.10e5 + 3.40e5i)T^{2} \) |
| 73 | \( 1 + (-216. + 333. i)T + (-1.58e5 - 3.55e5i)T^{2} \) |
| 79 | \( 1 + (411. + 925. i)T + (-3.29e5 + 3.66e5i)T^{2} \) |
| 83 | \( 1 + (73.2 - 462. i)T + (-5.43e5 - 1.76e5i)T^{2} \) |
| 89 | \( 1 + (-328. - 364. i)T + (-7.36e4 + 7.01e5i)T^{2} \) |
| 97 | \( 1 + (-83.1 - 524. i)T + (-8.68e5 + 2.82e5i)T^{2} \) |
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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
−11.80578604442349709788429413218, −10.97558977889748539371305861793, −10.52537121375023669248661143426, −9.086727990714108234426655248520, −8.728404723691379224173631853047, −8.051507080382249264236091742162, −6.79453793544935949794940451257, −4.36595316285152382771143626458, −3.42602419950622478736260337655, −1.50671022065758803929657938531,
0.932390455863758677552950238835, 1.79451606203804622517188300616, 3.55153968590262628846538162774, 6.31046082162699717330461564617, 7.43820277285533410402696230501, 7.988535359315447014482150460998, 8.647155418130342840488174997367, 9.368322944722939701654198024688, 11.21452305077957462067027484365, 11.57390747946365004935016644545
