L(s) = 1 | + (0.707 + 0.707i)2-s + (0.407 + 1.68i)3-s + 1.00i·4-s + (−0.707 + 0.707i)5-s + (−0.902 + 1.47i)6-s + 2.94·7-s + (−0.707 + 0.707i)8-s + (−2.66 + 1.37i)9-s − 1.00·10-s + 5.04·11-s + (−1.68 + 0.407i)12-s + (2.73 + 2.73i)13-s + (2.07 + 2.07i)14-s + (−1.47 − 0.902i)15-s − 1.00·16-s + (5.09 − 5.09i)17-s + ⋯ |
L(s) = 1 | + (0.499 + 0.499i)2-s + (0.235 + 0.971i)3-s + 0.500i·4-s + (−0.316 + 0.316i)5-s + (−0.368 + 0.603i)6-s + 1.11·7-s + (−0.250 + 0.250i)8-s + (−0.889 + 0.457i)9-s − 0.316·10-s + 1.52·11-s + (−0.485 + 0.117i)12-s + (0.758 + 0.758i)13-s + (0.555 + 0.555i)14-s + (−0.381 − 0.232i)15-s − 0.250·16-s + (1.23 − 1.23i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.578 - 0.815i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.578 - 0.815i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.621470452\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.621470452\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.707 - 0.707i)T \) |
| 3 | \( 1 + (-0.407 - 1.68i)T \) |
| 5 | \( 1 + (0.707 - 0.707i)T \) |
| 37 | \( 1 + (5.76 + 1.93i)T \) |
good | 7 | \( 1 - 2.94T + 7T^{2} \) |
| 11 | \( 1 - 5.04T + 11T^{2} \) |
| 13 | \( 1 + (-2.73 - 2.73i)T + 13iT^{2} \) |
| 17 | \( 1 + (-5.09 + 5.09i)T - 17iT^{2} \) |
| 19 | \( 1 + (-2.15 - 2.15i)T + 19iT^{2} \) |
| 23 | \( 1 + (3.53 - 3.53i)T - 23iT^{2} \) |
| 29 | \( 1 + (4.90 + 4.90i)T + 29iT^{2} \) |
| 31 | \( 1 + (4.78 - 4.78i)T - 31iT^{2} \) |
| 41 | \( 1 + 5.08T + 41T^{2} \) |
| 43 | \( 1 + (1.16 + 1.16i)T + 43iT^{2} \) |
| 47 | \( 1 + 6.36iT - 47T^{2} \) |
| 53 | \( 1 + 3.67iT - 53T^{2} \) |
| 59 | \( 1 + (-6.34 + 6.34i)T - 59iT^{2} \) |
| 61 | \( 1 + (-3.36 + 3.36i)T - 61iT^{2} \) |
| 67 | \( 1 - 11.8iT - 67T^{2} \) |
| 71 | \( 1 + 6.94iT - 71T^{2} \) |
| 73 | \( 1 + 8.41iT - 73T^{2} \) |
| 79 | \( 1 + (-10.3 - 10.3i)T + 79iT^{2} \) |
| 83 | \( 1 + 4.78iT - 83T^{2} \) |
| 89 | \( 1 + (-2.54 - 2.54i)T + 89iT^{2} \) |
| 97 | \( 1 + (9.97 + 9.97i)T + 97iT^{2} \) |
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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
−9.975898195353573946784461872407, −9.250373377689277176813147396633, −8.460302257741486006317509726816, −7.67576503376804091185528881285, −6.80012718860335791914701334884, −5.63670568024779571732016779830, −4.99737946054271160397845957758, −3.79067790797358212898603451882, −3.59552186972809320780993050654, −1.78177770465549232305953626837,
1.13869933528713732756907791407, 1.75573937420650848696102098969, 3.34507393198067855669256292480, 4.03957425763893693584768369660, 5.35686063792088269242768039821, 6.03547091682887076796313470678, 7.06590189074225946087617499068, 8.040898607581573461099367939380, 8.546418709813412789649020669840, 9.458930404935749813920886293274