| L(s) = 1 | + i·3-s + i·7-s + 2·9-s − 3·11-s − 4i·13-s + 4·19-s − 21-s + 5i·27-s + 2·31-s − 3i·33-s − i·37-s + 4·39-s + 3·41-s + 2i·43-s − 3i·47-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s + 0.377i·7-s + 0.666·9-s − 0.904·11-s − 1.10i·13-s + 0.917·19-s − 0.218·21-s + 0.962i·27-s + 0.359·31-s − 0.522i·33-s − 0.164i·37-s + 0.640·39-s + 0.468·41-s + 0.304i·43-s − 0.437i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.926122172\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.926122172\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 37 | \( 1 + iT \) |
| good | 3 | \( 1 - iT - 3T^{2} \) |
| 7 | \( 1 - iT - 7T^{2} \) |
| 11 | \( 1 + 3T + 11T^{2} \) |
| 13 | \( 1 + 4iT - 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 2T + 31T^{2} \) |
| 41 | \( 1 - 3T + 41T^{2} \) |
| 43 | \( 1 - 2iT - 43T^{2} \) |
| 47 | \( 1 + 3iT - 47T^{2} \) |
| 53 | \( 1 + 9iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 - 4iT - 67T^{2} \) |
| 71 | \( 1 - 15T + 71T^{2} \) |
| 73 | \( 1 + 7iT - 73T^{2} \) |
| 79 | \( 1 - 10T + 79T^{2} \) |
| 83 | \( 1 + 3iT - 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 - 10iT - 97T^{2} \) |
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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
−8.554270329015653020170489075141, −7.83608103053543216222784333625, −7.26787379253109352729481592502, −6.26368128535285969529855408832, −5.30592764654037097023319160359, −5.01817260584901549641342378667, −3.89520372417640665920409716266, −3.13595360159329522487110566349, −2.21574578173145753497298796221, −0.810056413509317160006247661139,
0.837451010560214352325479717699, 1.85578817883024117628056677975, 2.78018821325876892007237404289, 3.91547116443758831253010757908, 4.61875980721534448315406222954, 5.49421923873996548384110749761, 6.41262034684198121545810791204, 7.10447300679373684773382516344, 7.59711125998413807505387237226, 8.298986268805390454116543866095
