L(s) = 1 | + i·3-s + 4.05·7-s − 9-s + 0.985i·11-s + 4.94i·13-s + 4.52·17-s − 2.60i·19-s + 4.05i·21-s − 3.53·23-s − i·27-s + 7.59i·29-s + 3.28·31-s − 0.985·33-s + 0.945i·37-s − 4.94·39-s + ⋯ |
L(s) = 1 | + 0.577i·3-s + 1.53·7-s − 0.333·9-s + 0.297i·11-s + 1.37i·13-s + 1.09·17-s − 0.597i·19-s + 0.885i·21-s − 0.737·23-s − 0.192i·27-s + 1.41i·29-s + 0.589·31-s − 0.171·33-s + 0.155i·37-s − 0.791·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.227 - 0.973i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.227 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.135096277\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.135096277\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 4.05T + 7T^{2} \) |
| 11 | \( 1 - 0.985iT - 11T^{2} \) |
| 13 | \( 1 - 4.94iT - 13T^{2} \) |
| 17 | \( 1 - 4.52T + 17T^{2} \) |
| 19 | \( 1 + 2.60iT - 19T^{2} \) |
| 23 | \( 1 + 3.53T + 23T^{2} \) |
| 29 | \( 1 - 7.59iT - 29T^{2} \) |
| 31 | \( 1 - 3.28T + 31T^{2} \) |
| 37 | \( 1 - 0.945iT - 37T^{2} \) |
| 41 | \( 1 - 0.568T + 41T^{2} \) |
| 43 | \( 1 + 8.45iT - 43T^{2} \) |
| 47 | \( 1 + 2.60T + 47T^{2} \) |
| 53 | \( 1 - 0.229iT - 53T^{2} \) |
| 59 | \( 1 - 9.10iT - 59T^{2} \) |
| 61 | \( 1 - 11.0iT - 61T^{2} \) |
| 67 | \( 1 + 8.45iT - 67T^{2} \) |
| 71 | \( 1 + 1.43T + 71T^{2} \) |
| 73 | \( 1 - 11.9T + 73T^{2} \) |
| 79 | \( 1 - 3.28T + 79T^{2} \) |
| 83 | \( 1 + 9.89iT - 83T^{2} \) |
| 89 | \( 1 + 12.3T + 89T^{2} \) |
| 97 | \( 1 + 3.23T + 97T^{2} \) |
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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
−9.009760401162835505489099544198, −8.499604845821540504325876127630, −7.62714495183199757698383277130, −6.94800365289744374429676518598, −5.84087035123128219109016276869, −4.97835745588740885559111085652, −4.48802128083501756787753165583, −3.57730477266608480332732201678, −2.27533373542422783659852640126, −1.33901661315084443447716664787,
0.797383733837964632027325876687, 1.78832548450302579599134299785, 2.86121457229029231920499736410, 3.92922141651802916002986070981, 5.01229810573226381218438696412, 5.63893305035678295063850175710, 6.37509698357040722752428404339, 7.65416445226518651447963941911, 8.030822397355998048119246126364, 8.311506654261650652435819968158