L(s) = 1 | + 2.56i·2-s + 3.68i·3-s + 1.43·4-s − 9.43·6-s + 18.1i·7-s + 24.1i·8-s + 13.4·9-s + 64.7·11-s + 5.30i·12-s + 13i·13-s − 46.5·14-s − 50.4·16-s − 25.5i·17-s + 34.3i·18-s + 107.·19-s + ⋯ |
L(s) = 1 | + 0.905i·2-s + 0.709i·3-s + 0.179·4-s − 0.642·6-s + 0.981i·7-s + 1.06i·8-s + 0.497·9-s + 1.77·11-s + 0.127i·12-s + 0.277i·13-s − 0.888·14-s − 0.787·16-s − 0.364i·17-s + 0.450i·18-s + 1.30·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(2.479952775\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.479952775\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 13 | \( 1 - 13iT \) |
good | 2 | \( 1 - 2.56iT - 8T^{2} \) |
| 3 | \( 1 - 3.68iT - 27T^{2} \) |
| 7 | \( 1 - 18.1iT - 343T^{2} \) |
| 11 | \( 1 - 64.7T + 1.33e3T^{2} \) |
| 17 | \( 1 + 25.5iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 107.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 73.2iT - 1.21e4T^{2} \) |
| 29 | \( 1 + 175.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 113.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 114. iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 69.6T + 6.89e4T^{2} \) |
| 43 | \( 1 + 438. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 31.9iT - 1.03e5T^{2} \) |
| 53 | \( 1 + 2.84iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 71.6T + 2.05e5T^{2} \) |
| 61 | \( 1 + 920.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 444. iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 541.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 764. iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 421.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 603. iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 1.15e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 583. iT - 9.12e5T^{2} \) |
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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
−11.72136503609301718177513161390, −10.62753553101702144082288535148, −9.249783731033317020308087182306, −9.040667915827122377056101448542, −7.56961256098435173752296635982, −6.71758275121854846080532643478, −5.76390430970749247955223197594, −4.76779479718188982490420179830, −3.46048721376878680794274874190, −1.78253068736606955519741748194,
1.01438531445432514586042981298, 1.64359821011770259688267907668, 3.39846612707465757554114803862, 4.19353380197353844081808288892, 6.08596167276612648476747002950, 7.08741802148926598695932822741, 7.56839771430920734485331135983, 9.273423850468515179954713811614, 9.931809113758408879000518478340, 11.00725374196556733210250762955