| L(s) = 1 | − 2i·2-s + 2i·3-s − 4·4-s + 4·6-s − 31i·7-s + 8i·8-s + 23·9-s + 57·11-s − 8i·12-s + 52i·13-s − 62·14-s + 16·16-s + 69i·17-s − 46i·18-s − 19·19-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s + 0.384i·3-s − 0.5·4-s + 0.272·6-s − 1.67i·7-s + 0.353i·8-s + 0.851·9-s + 1.56·11-s − 0.192i·12-s + 1.10i·13-s − 1.18·14-s + 0.250·16-s + 0.984i·17-s − 0.602i·18-s − 0.229·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.363737575\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.363737575\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2iT \) |
| 5 | \( 1 \) |
| 19 | \( 1 + 19T \) |
| good | 3 | \( 1 - 2iT - 27T^{2} \) |
| 7 | \( 1 + 31iT - 343T^{2} \) |
| 11 | \( 1 - 57T + 1.33e3T^{2} \) |
| 13 | \( 1 - 52iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 69iT - 4.91e3T^{2} \) |
| 23 | \( 1 - 72iT - 1.21e4T^{2} \) |
| 29 | \( 1 - 150T + 2.43e4T^{2} \) |
| 31 | \( 1 - 32T + 2.97e4T^{2} \) |
| 37 | \( 1 + 226iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 258T + 6.89e4T^{2} \) |
| 43 | \( 1 - 67iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 579iT - 1.03e5T^{2} \) |
| 53 | \( 1 - 432iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 330T + 2.05e5T^{2} \) |
| 61 | \( 1 + 13T + 2.26e5T^{2} \) |
| 67 | \( 1 + 856iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 642T + 3.57e5T^{2} \) |
| 73 | \( 1 - 487iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 700T + 4.93e5T^{2} \) |
| 83 | \( 1 - 12iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 600T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.42e3iT - 9.12e5T^{2} \) |
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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
−9.644064037140311240561006068300, −9.161236716932795085658509517337, −7.957922448707580277618440977216, −6.97215459298714555403848428555, −6.36735309107665145319883590926, −4.65555582243016300614076263349, −4.07776084083849348591990500347, −3.60414352446249723225262212765, −1.70041086502442814865661357666, −1.02200080965530189226789347875,
0.78135078342163339690231969317, 2.13248276283789345642119631744, 3.37379483734888320502834456982, 4.67786630321155019324172541633, 5.48114803641410353242478146921, 6.48890752218509892622696000315, 6.91113085207857680330353429048, 8.194347193438798745591987146293, 8.673752622207650673948618893327, 9.546128984383800734137750149014
