| L(s) = 1 | + 3·9-s − 2·11-s + 2i·13-s + 2·19-s + 8i·23-s − 29-s + 2·31-s − 4i·37-s − 10·41-s − 4i·43-s + 12i·47-s + 7·49-s + 6i·53-s + 12·59-s − 10·61-s + ⋯ |
| L(s) = 1 | + 9-s − 0.603·11-s + 0.554i·13-s + 0.458·19-s + 1.66i·23-s − 0.185·29-s + 0.359·31-s − 0.657i·37-s − 1.56·41-s − 0.609i·43-s + 1.75i·47-s + 49-s + 0.824i·53-s + 1.56·59-s − 1.28·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.704403235\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.704403235\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 29 | \( 1 + T \) |
| good | 3 | \( 1 - 3T^{2} \) |
| 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 - 2iT - 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 - 2T + 19T^{2} \) |
| 23 | \( 1 - 8iT - 23T^{2} \) |
| 31 | \( 1 - 2T + 31T^{2} \) |
| 37 | \( 1 + 4iT - 37T^{2} \) |
| 41 | \( 1 + 10T + 41T^{2} \) |
| 43 | \( 1 + 4iT - 43T^{2} \) |
| 47 | \( 1 - 12iT - 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 - 12T + 59T^{2} \) |
| 61 | \( 1 + 10T + 61T^{2} \) |
| 67 | \( 1 - 12iT - 67T^{2} \) |
| 71 | \( 1 - 12T + 71T^{2} \) |
| 73 | \( 1 + 12iT - 73T^{2} \) |
| 79 | \( 1 + 2T + 79T^{2} \) |
| 83 | \( 1 - 4iT - 83T^{2} \) |
| 89 | \( 1 - 10T + 89T^{2} \) |
| 97 | \( 1 - 8iT - 97T^{2} \) |
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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
−9.052978364136340422138182232645, −7.981080650961969339112582302446, −7.40905058891638774434677971101, −6.77931181155221894837203349012, −5.76455817118344703199740287192, −5.04591397138533902146460590538, −4.15731252643609686278314349118, −3.36132623144193292438649194868, −2.18101251717164426773249743024, −1.18601639715493187244629271629,
0.59435553238787811240763273639, 1.90406125190320875997852166940, 2.92133659661270515027024579327, 3.88751859682147661393167948392, 4.80493460468917129426563953448, 5.41279940944279165696937535638, 6.54379005075471290013237555818, 7.02680600150139887846831327106, 8.000137497253049560981281616718, 8.453442722646973244918741045268
