| L(s) = 1 | − 1.53i·2-s + i·3-s − 0.361·4-s + (1.20 + 1.88i)5-s + 1.53·6-s − 2.86i·7-s − 2.51i·8-s − 9-s + (2.89 − 1.85i)10-s − 0.361i·12-s − 2.43i·13-s − 4.40·14-s + (−1.88 + 1.20i)15-s − 4.59·16-s − 3.88i·17-s + 1.53i·18-s + ⋯ |
| L(s) = 1 | − 1.08i·2-s + 0.577i·3-s − 0.180·4-s + (0.540 + 0.841i)5-s + 0.627·6-s − 1.08i·7-s − 0.890i·8-s − 0.333·9-s + (0.914 − 0.587i)10-s − 0.104i·12-s − 0.675i·13-s − 1.17·14-s + (−0.485 + 0.311i)15-s − 1.14·16-s − 0.942i·17-s + 0.362i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.841 + 0.540i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.841 + 0.540i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.515599050\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.515599050\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-1.20 - 1.88i)T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 1.53iT - 2T^{2} \) |
| 7 | \( 1 + 2.86iT - 7T^{2} \) |
| 13 | \( 1 + 2.43iT - 13T^{2} \) |
| 17 | \( 1 + 3.88iT - 17T^{2} \) |
| 19 | \( 1 + 5.93T + 19T^{2} \) |
| 23 | \( 1 + 6.02iT - 23T^{2} \) |
| 29 | \( 1 + 4.03T + 29T^{2} \) |
| 31 | \( 1 + 0.783T + 31T^{2} \) |
| 37 | \( 1 + 8.02iT - 37T^{2} \) |
| 41 | \( 1 + 7.54T + 41T^{2} \) |
| 43 | \( 1 - 9.90iT - 43T^{2} \) |
| 47 | \( 1 + 0.551iT - 47T^{2} \) |
| 53 | \( 1 - 2.75iT - 53T^{2} \) |
| 59 | \( 1 - 14.5T + 59T^{2} \) |
| 61 | \( 1 - 13.7T + 61T^{2} \) |
| 67 | \( 1 + 3.96iT - 67T^{2} \) |
| 71 | \( 1 - 9.27T + 71T^{2} \) |
| 73 | \( 1 - 0.615iT - 73T^{2} \) |
| 79 | \( 1 + 4.53T + 79T^{2} \) |
| 83 | \( 1 + 11.6iT - 83T^{2} \) |
| 89 | \( 1 + 16.9T + 89T^{2} \) |
| 97 | \( 1 + 8.95iT - 97T^{2} \) |
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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
−9.378701981995552231235597379356, −8.312642481555407761606257707987, −7.15755088550838953347819699121, −6.70377200033657047678070038310, −5.65497108514272075240728281119, −4.45070344632794348671804778345, −3.72885383551585368147776370231, −2.86408332637609515346180623077, −2.05610498247138737572360857775, −0.50958314540737450895364198970,
1.74016947945451215652759386177, 2.29512991629579920112085808978, 3.97099057156659495705649293601, 5.28879170993574105089421772003, 5.57395263780869323079310616611, 6.46511231881179074209711694557, 6.98707886489600596739174408568, 8.305989953510412090009755318549, 8.406620391359648443316318259951, 9.185515110841463596367076899687
