| L(s) = 1 | + (−0.470 + 0.0619i)2-s + (−0.749 − 1.51i)3-s + (−1.71 + 0.459i)4-s + (0.751 − 0.659i)5-s + (0.446 + 0.668i)6-s + (0.515 + 2.59i)7-s + (1.65 − 0.685i)8-s + (0.0798 − 0.104i)9-s + (−0.312 + 0.356i)10-s + (0.241 − 0.0158i)11-s + (1.98 + 2.26i)12-s + (−0.876 − 0.876i)13-s + (−0.403 − 1.18i)14-s + (−1.56 − 0.648i)15-s + (2.33 − 1.35i)16-s + (2.47 − 3.30i)17-s + ⋯ |
| L(s) = 1 | + (−0.332 + 0.0437i)2-s + (−0.432 − 0.877i)3-s + (−0.857 + 0.229i)4-s + (0.336 − 0.294i)5-s + (0.182 + 0.272i)6-s + (0.194 + 0.980i)7-s + (0.584 − 0.242i)8-s + (0.0266 − 0.0346i)9-s + (−0.0989 + 0.112i)10-s + (0.0728 − 0.00477i)11-s + (0.572 + 0.652i)12-s + (−0.242 − 0.242i)13-s + (−0.107 − 0.317i)14-s + (−0.404 − 0.167i)15-s + (0.584 − 0.337i)16-s + (0.599 − 0.800i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 595 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0288 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 595 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0288 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.615373 - 0.597851i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.615373 - 0.597851i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-0.751 + 0.659i)T \) |
| 7 | \( 1 + (-0.515 - 2.59i)T \) |
| 17 | \( 1 + (-2.47 + 3.30i)T \) |
| good | 2 | \( 1 + (0.470 - 0.0619i)T + (1.93 - 0.517i)T^{2} \) |
| 3 | \( 1 + (0.749 + 1.51i)T + (-1.82 + 2.38i)T^{2} \) |
| 11 | \( 1 + (-0.241 + 0.0158i)T + (10.9 - 1.43i)T^{2} \) |
| 13 | \( 1 + (0.876 + 0.876i)T + 13iT^{2} \) |
| 19 | \( 1 + (-0.384 - 2.92i)T + (-18.3 + 4.91i)T^{2} \) |
| 23 | \( 1 + (-1.34 - 0.664i)T + (14.0 + 18.2i)T^{2} \) |
| 29 | \( 1 + (1.61 + 8.09i)T + (-26.7 + 11.0i)T^{2} \) |
| 31 | \( 1 + (-3.30 + 1.62i)T + (18.8 - 24.5i)T^{2} \) |
| 37 | \( 1 + (0.0712 - 1.08i)T + (-36.6 - 4.82i)T^{2} \) |
| 41 | \( 1 + (-2.20 + 11.0i)T + (-37.8 - 15.6i)T^{2} \) |
| 43 | \( 1 + (4.59 + 11.1i)T + (-30.4 + 30.4i)T^{2} \) |
| 47 | \( 1 + (-1.12 + 4.19i)T + (-40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (8.77 + 11.4i)T + (-13.7 + 51.1i)T^{2} \) |
| 59 | \( 1 + (-13.4 - 1.77i)T + (56.9 + 15.2i)T^{2} \) |
| 61 | \( 1 + (-4.86 - 14.3i)T + (-48.3 + 37.1i)T^{2} \) |
| 67 | \( 1 + (-4.07 - 2.35i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-4.23 - 2.82i)T + (27.1 + 65.5i)T^{2} \) |
| 73 | \( 1 + (6.08 + 2.06i)T + (57.9 + 44.4i)T^{2} \) |
| 79 | \( 1 + (-6.08 + 12.3i)T + (-48.0 - 62.6i)T^{2} \) |
| 83 | \( 1 + (4.57 - 11.0i)T + (-58.6 - 58.6i)T^{2} \) |
| 89 | \( 1 + (-11.8 - 3.17i)T + (77.0 + 44.5i)T^{2} \) |
| 97 | \( 1 + (4.44 - 0.884i)T + (89.6 - 37.1i)T^{2} \) |
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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
−10.16535121968763227912464879082, −9.549228170673769969591258807047, −8.667059240400533164744317547039, −7.888887909534951215577225567664, −6.99674918775626411903223283576, −5.76356260872826254351907868028, −5.19104385189161491102791750788, −3.79473240225137167157354505365, −2.13711087093297916558105702571, −0.66779816329875487775820147396,
1.34255559969133805715502100801, 3.43524090309747255504231066819, 4.55006643417275095195150463029, 5.02659259568890343679009599281, 6.28959708181767616710907819831, 7.45929320198344696701700233261, 8.366569379283839708033122263195, 9.576280155657562273553814224119, 9.888174930934803461688301308671, 10.82086399415744618797083485192
