| L(s) = 1 | + i·2-s + i·3-s − 4-s + (2 − i)5-s − 6-s + 4i·7-s − i·8-s − 9-s + (1 + 2i)10-s + 2·11-s − i·12-s − i·13-s − 4·14-s + (1 + 2i)15-s + 16-s + 4i·17-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s + 0.577i·3-s − 0.5·4-s + (0.894 − 0.447i)5-s − 0.408·6-s + 1.51i·7-s − 0.353i·8-s − 0.333·9-s + (0.316 + 0.632i)10-s + 0.603·11-s − 0.288i·12-s − 0.277i·13-s − 1.06·14-s + (0.258 + 0.516i)15-s + 0.250·16-s + 0.970i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.763100 + 1.23472i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.763100 + 1.23472i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - iT \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 + (-2 + i)T \) |
| 13 | \( 1 + iT \) |
| good | 7 | \( 1 - 4iT - 7T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 17 | \( 1 - 4iT - 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 23 | \( 1 - 6iT - 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 + 6iT - 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 + 8iT - 43T^{2} \) |
| 47 | \( 1 - 8iT - 47T^{2} \) |
| 53 | \( 1 + 10iT - 53T^{2} \) |
| 59 | \( 1 - 14T + 59T^{2} \) |
| 61 | \( 1 - 10T + 61T^{2} \) |
| 67 | \( 1 - 4iT - 67T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 + 10iT - 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 + 12iT - 83T^{2} \) |
| 89 | \( 1 - 18T + 89T^{2} \) |
| 97 | \( 1 - 6iT - 97T^{2} \) |
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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
−11.70541008195990028035299899017, −10.44950589932283994661408709602, −9.475364833362792739908843742432, −8.927157072855751300511983482411, −8.199621978548240832095296968422, −6.59800822985621947604709322250, −5.69097944702842112330053025984, −5.19510855767203085831627464049, −3.72794716259632351621374002454, −2.04586530366640465688845444023,
1.06042640914004630859042630322, 2.44101128541060251824225185472, 3.78286145293792027194094329732, 4.99600095108582412245908631971, 6.52567964516979694630917506526, 7.02555179057765223695230575436, 8.323560979338947872027091728120, 9.453993206689234124581425350749, 10.23845136228507693970389687210, 10.95030909764592178474522169338
