L(s) = 1 | + (−0.5 + 0.866i)5-s + (0.5 − 2.59i)7-s + (1.5 + 2.59i)11-s + 4·13-s + (2 − 3.46i)19-s + (4 − 6.92i)23-s + (2 + 3.46i)25-s + 3·29-s + (2.5 + 4.33i)31-s + (2 + 1.73i)35-s + (−4 + 6.92i)37-s − 8·41-s + 6·43-s + (5 − 8.66i)47-s + (−6.5 − 2.59i)49-s + ⋯ |
L(s) = 1 | + (−0.223 + 0.387i)5-s + (0.188 − 0.981i)7-s + (0.452 + 0.783i)11-s + 1.10·13-s + (0.458 − 0.794i)19-s + (0.834 − 1.44i)23-s + (0.400 + 0.692i)25-s + 0.557·29-s + (0.449 + 0.777i)31-s + (0.338 + 0.292i)35-s + (−0.657 + 1.13i)37-s − 1.24·41-s + 0.914·43-s + (0.729 − 1.26i)47-s + (−0.928 − 0.371i)49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 + 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.50702 - 0.0956360i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.50702 - 0.0956360i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-0.5 + 2.59i)T \) |
good | 5 | \( 1 + (0.5 - 0.866i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-1.5 - 2.59i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 4T + 13T^{2} \) |
| 17 | \( 1 + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-2 + 3.46i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-4 + 6.92i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 3T + 29T^{2} \) |
| 31 | \( 1 + (-2.5 - 4.33i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (4 - 6.92i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 8T + 41T^{2} \) |
| 43 | \( 1 - 6T + 43T^{2} \) |
| 47 | \( 1 + (-5 + 8.66i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-4.5 - 7.79i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (2.5 + 4.33i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-5 + 8.66i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (3 + 5.19i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 10T + 71T^{2} \) |
| 73 | \( 1 + (1 + 1.73i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (5.5 - 9.52i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 7T + 83T^{2} \) |
| 89 | \( 1 + (9 - 15.5i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 17T + 97T^{2} \) |
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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
−10.79456251038982872073303338742, −10.25835875651490948168694898395, −9.065068956014744154161589686847, −8.236893518893041840682346735070, −6.96023274190169157949048044455, −6.72930363099288954951121516728, −5.07057680900019599616935491657, −4.13431469008601160659979963729, −3.02037430562676103778859869525, −1.21089169785677624677076404699,
1.32712485904019117575690321372, 3.01535134390573336343941041426, 4.12653489596136534809295839707, 5.50403315614617341918933317591, 6.06227841437955689729721380516, 7.39915350766115575884578249705, 8.573070825867855166115179036695, 8.840257192656848746535689843757, 10.00263326156933108535713621077, 11.16629336353761874326569095686