L(s) = 1 | − 2i·3-s − 9-s − 4·11-s + 4i·13-s + 2i·17-s − 19-s − 4i·23-s − 4i·27-s + 6·29-s − 8·31-s + 8i·33-s + 4i·37-s + 8·39-s − 2·41-s + 4i·43-s + ⋯ |
L(s) = 1 | − 1.15i·3-s − 0.333·9-s − 1.20·11-s + 1.10i·13-s + 0.485i·17-s − 0.229·19-s − 0.834i·23-s − 0.769i·27-s + 1.11·29-s − 1.43·31-s + 1.39i·33-s + 0.657i·37-s + 1.28·39-s − 0.312·41-s + 0.609i·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.250727010\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.250727010\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 + T \) |
good | 3 | \( 1 + 2iT - 3T^{2} \) |
| 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 + 4T + 11T^{2} \) |
| 13 | \( 1 - 4iT - 13T^{2} \) |
| 17 | \( 1 - 2iT - 17T^{2} \) |
| 23 | \( 1 + 4iT - 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + 8T + 31T^{2} \) |
| 37 | \( 1 - 4iT - 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 - 8iT - 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 - 8T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 - 14iT - 67T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 - 6iT - 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 - 16iT - 83T^{2} \) |
| 89 | \( 1 - 18T + 89T^{2} \) |
| 97 | \( 1 - 4iT - 97T^{2} \) |
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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
−8.368594485502377344560929485343, −7.79429008931522625111058771492, −7.03026440298664264605205141397, −6.53129176987406667258668231547, −5.74021814745134973998248861979, −4.79337381446210053928085388874, −3.99008239393249547241255919867, −2.70418186520687253034719450080, −2.06889194092672978034692285573, −1.02663657397353437532918763041,
0.40693001706152089466805959728, 2.09514546165216426893649880899, 3.15642961271282818174487236214, 3.72339528297593534327338333426, 4.81891976734193592567706230130, 5.25087172880662677657622445678, 5.92802227178778727402680733423, 7.19083435255323682069450919347, 7.67473378577160993432617844906, 8.600490612133485140072493170118