| L(s) = 1 | + 2.25i·2-s − i·3-s − 3.09·4-s + 2.25·6-s + 1.55i·7-s − 2.48i·8-s − 9-s − 3.75·11-s + 3.09i·12-s + 6.79i·13-s − 3.51·14-s − 0.596·16-s + 3.85i·17-s − 2.25i·18-s + 2.13·19-s + ⋯ |
| L(s) = 1 | + 1.59i·2-s − 0.577i·3-s − 1.54·4-s + 0.921·6-s + 0.587i·7-s − 0.876i·8-s − 0.333·9-s − 1.13·11-s + 0.894i·12-s + 1.88i·13-s − 0.938·14-s − 0.149·16-s + 0.934i·17-s − 0.532i·18-s + 0.489·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 375 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 375 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(-0.954097i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(-0.954097i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + iT \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - 2.25iT - 2T^{2} \) |
| 7 | \( 1 - 1.55iT - 7T^{2} \) |
| 11 | \( 1 + 3.75T + 11T^{2} \) |
| 13 | \( 1 - 6.79iT - 13T^{2} \) |
| 17 | \( 1 - 3.85iT - 17T^{2} \) |
| 19 | \( 1 - 2.13T + 19T^{2} \) |
| 23 | \( 1 + 0.420iT - 23T^{2} \) |
| 29 | \( 1 + 9.30T + 29T^{2} \) |
| 31 | \( 1 - 9.92T + 31T^{2} \) |
| 37 | \( 1 + 6.83iT - 37T^{2} \) |
| 41 | \( 1 - 5.68T + 41T^{2} \) |
| 43 | \( 1 - 3.59iT - 43T^{2} \) |
| 47 | \( 1 + 4.13iT - 47T^{2} \) |
| 53 | \( 1 - 6.72iT - 53T^{2} \) |
| 59 | \( 1 - 3.19T + 59T^{2} \) |
| 61 | \( 1 + 1.49T + 61T^{2} \) |
| 67 | \( 1 - 1.47iT - 67T^{2} \) |
| 71 | \( 1 - 7.18T + 71T^{2} \) |
| 73 | \( 1 + 2.51iT - 73T^{2} \) |
| 79 | \( 1 + 2.00T + 79T^{2} \) |
| 83 | \( 1 - 2.60iT - 83T^{2} \) |
| 89 | \( 1 + 14.3T + 89T^{2} \) |
| 97 | \( 1 - 12.5iT - 97T^{2} \) |
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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
−11.97562932476310677865624421872, −11.00544568975414898165590251522, −9.524536147123492949150268583984, −8.744929932684760342434134069097, −7.889137650203649725369201060818, −7.11727174794117976510391503333, −6.19044106074516943103455665937, −5.45396582806554591054951966623, −4.25814270025030682342207605766, −2.23648002572703018875253857740,
0.63511139415157231085171383201, 2.65729844375092725379041936140, 3.39519526275468212979743315100, 4.67173347824719704141828490813, 5.56582487451345593329988336308, 7.42861391305520073277177089071, 8.378745425653071727909312586194, 9.695750374758327382491475529808, 10.13596192687297980175752021779, 10.86062922578029321671500178567
