L(s) = 1 | + 2.61·2-s + 2.91·3-s + 4.85·4-s + 7.63·6-s − 2.00·7-s + 7.46·8-s + 5.50·9-s − 1.47·11-s + 14.1·12-s + 0.967·13-s − 5.26·14-s + 9.83·16-s + 6.02·17-s + 14.4·18-s + 0.541·19-s − 5.86·21-s − 3.86·22-s − 1.49·23-s + 21.7·24-s + 2.53·26-s + 7.29·27-s − 9.75·28-s − 6.69·29-s + 6.96·31-s + 10.8·32-s − 4.30·33-s + 15.7·34-s + ⋯ |
L(s) = 1 | + 1.85·2-s + 1.68·3-s + 2.42·4-s + 3.11·6-s − 0.759·7-s + 2.63·8-s + 1.83·9-s − 0.445·11-s + 4.08·12-s + 0.268·13-s − 1.40·14-s + 2.45·16-s + 1.46·17-s + 3.39·18-s + 0.124·19-s − 1.27·21-s − 0.824·22-s − 0.311·23-s + 4.44·24-s + 0.496·26-s + 1.40·27-s − 1.84·28-s − 1.24·29-s + 1.25·31-s + 1.91·32-s − 0.749·33-s + 2.70·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(11.90365067\) |
\(L(\frac12)\) |
\(\approx\) |
\(11.90365067\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 241 | \( 1 - T \) |
good | 2 | \( 1 - 2.61T + 2T^{2} \) |
| 3 | \( 1 - 2.91T + 3T^{2} \) |
| 7 | \( 1 + 2.00T + 7T^{2} \) |
| 11 | \( 1 + 1.47T + 11T^{2} \) |
| 13 | \( 1 - 0.967T + 13T^{2} \) |
| 17 | \( 1 - 6.02T + 17T^{2} \) |
| 19 | \( 1 - 0.541T + 19T^{2} \) |
| 23 | \( 1 + 1.49T + 23T^{2} \) |
| 29 | \( 1 + 6.69T + 29T^{2} \) |
| 31 | \( 1 - 6.96T + 31T^{2} \) |
| 37 | \( 1 + 7.43T + 37T^{2} \) |
| 41 | \( 1 - 6.91T + 41T^{2} \) |
| 43 | \( 1 + 8.06T + 43T^{2} \) |
| 47 | \( 1 - 4.13T + 47T^{2} \) |
| 53 | \( 1 - 5.72T + 53T^{2} \) |
| 59 | \( 1 + 4.68T + 59T^{2} \) |
| 61 | \( 1 + 3.80T + 61T^{2} \) |
| 67 | \( 1 - 2.54T + 67T^{2} \) |
| 71 | \( 1 - 2.73T + 71T^{2} \) |
| 73 | \( 1 - 4.42T + 73T^{2} \) |
| 79 | \( 1 + 4.05T + 79T^{2} \) |
| 83 | \( 1 + 1.66T + 83T^{2} \) |
| 89 | \( 1 + 2.78T + 89T^{2} \) |
| 97 | \( 1 + 14.3T + 97T^{2} \) |
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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
−7.81885281578437359237747996828, −7.34765973828678450702062423289, −6.58049624456200095686494797173, −5.79859616459883094120095435735, −5.09967343398601653458740852804, −4.12448091948782585583097698817, −3.54519831031675664280494956282, −3.08263510998612060355984027407, −2.42857970518370921751504614059, −1.50324877070737559067333450660,
1.50324877070737559067333450660, 2.42857970518370921751504614059, 3.08263510998612060355984027407, 3.54519831031675664280494956282, 4.12448091948782585583097698817, 5.09967343398601653458740852804, 5.79859616459883094120095435735, 6.58049624456200095686494797173, 7.34765973828678450702062423289, 7.81885281578437359237747996828