| L(s) = 1 | − 4.09e3i·2-s + 7.01e5i·3-s − 1.67e7·4-s + 2.87e9·6-s − 7.10e8i·7-s + 6.87e10i·8-s + 3.54e11·9-s + 6.69e12·11-s − 1.17e13i·12-s + 4.06e12i·13-s − 2.90e12·14-s + 2.81e14·16-s + 2.05e15i·17-s − 1.45e15i·18-s − 1.32e16·19-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s + 0.762i·3-s − 0.5·4-s + 0.539·6-s − 0.0193i·7-s + 0.353i·8-s + 0.418·9-s + 0.642·11-s − 0.381i·12-s + 0.0483i·13-s − 0.0137·14-s + 0.250·16-s + 0.855i·17-s − 0.295i·18-s − 1.37·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(26-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s+25/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(13)\) |
\(\approx\) |
\(0.6075063248\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6075063248\) |
| \(L(\frac{27}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 4.09e3iT \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 7.01e5iT - 8.47e11T^{2} \) |
| 7 | \( 1 + 7.10e8iT - 1.34e21T^{2} \) |
| 11 | \( 1 - 6.69e12T + 1.08e26T^{2} \) |
| 13 | \( 1 - 4.06e12iT - 7.05e27T^{2} \) |
| 17 | \( 1 - 2.05e15iT - 5.77e30T^{2} \) |
| 19 | \( 1 + 1.32e16T + 9.30e31T^{2} \) |
| 23 | \( 1 - 2.83e16iT - 1.10e34T^{2} \) |
| 29 | \( 1 + 8.79e17T + 3.63e36T^{2} \) |
| 31 | \( 1 + 3.20e17T + 1.92e37T^{2} \) |
| 37 | \( 1 + 4.64e19iT - 1.60e39T^{2} \) |
| 41 | \( 1 + 1.18e20T + 2.08e40T^{2} \) |
| 43 | \( 1 - 1.34e20iT - 6.86e40T^{2} \) |
| 47 | \( 1 + 7.78e20iT - 6.34e41T^{2} \) |
| 53 | \( 1 - 3.18e21iT - 1.27e43T^{2} \) |
| 59 | \( 1 - 2.03e21T + 1.86e44T^{2} \) |
| 61 | \( 1 - 3.16e22T + 4.29e44T^{2} \) |
| 67 | \( 1 - 4.98e22iT - 4.48e45T^{2} \) |
| 71 | \( 1 - 1.16e23T + 1.91e46T^{2} \) |
| 73 | \( 1 - 1.07e23iT - 3.82e46T^{2} \) |
| 79 | \( 1 + 2.51e23T + 2.75e47T^{2} \) |
| 83 | \( 1 - 1.47e24iT - 9.48e47T^{2} \) |
| 89 | \( 1 - 1.69e24T + 5.42e48T^{2} \) |
| 97 | \( 1 + 4.56e24iT - 4.66e49T^{2} \) |
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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
−11.09073298684597488337133826814, −10.32573291482064191219288769677, −9.369702598308829097081130191966, −8.409988587590998819210643331778, −6.86967683782969449094406017182, −5.50346991322674737867307985902, −4.19711503533296956641994840307, −3.75681242092132144833573822475, −2.24813025760978642019434383754, −1.25519556630877727280724094134,
0.11585473837511834244886636408, 1.15662445306369396906185232805, 2.27628633879304211092293677414, 3.83475545778180951225738188826, 4.92437090223178012415382602729, 6.31513769984505748271289946227, 6.94941453932342488415563784546, 8.000173904957022883862724517986, 9.051434958784286946095484911092, 10.24143328178033681233264652161
