L(s) = 1 | − 1.96·2-s − 3-s + 1.85·4-s − 5-s + 1.96·6-s + 7-s + 0.289·8-s + 9-s + 1.96·10-s − 1.88·11-s − 1.85·12-s − 6.73·13-s − 1.96·14-s + 15-s − 4.27·16-s + 7.00·17-s − 1.96·18-s + 2.74·19-s − 1.85·20-s − 21-s + 3.69·22-s − 23-s − 0.289·24-s + 25-s + 13.2·26-s − 27-s + 1.85·28-s + ⋯ |
L(s) = 1 | − 1.38·2-s − 0.577·3-s + 0.926·4-s − 0.447·5-s + 0.801·6-s + 0.377·7-s + 0.102·8-s + 0.333·9-s + 0.620·10-s − 0.567·11-s − 0.534·12-s − 1.86·13-s − 0.524·14-s + 0.258·15-s − 1.06·16-s + 1.70·17-s − 0.462·18-s + 0.629·19-s − 0.414·20-s − 0.218·21-s + 0.787·22-s − 0.208·23-s − 0.0591·24-s + 0.200·25-s + 2.59·26-s − 0.192·27-s + 0.350·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2415 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2415 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4282732293\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4282732293\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 + T \) |
good | 2 | \( 1 + 1.96T + 2T^{2} \) |
| 11 | \( 1 + 1.88T + 11T^{2} \) |
| 13 | \( 1 + 6.73T + 13T^{2} \) |
| 17 | \( 1 - 7.00T + 17T^{2} \) |
| 19 | \( 1 - 2.74T + 19T^{2} \) |
| 29 | \( 1 + 4.49T + 29T^{2} \) |
| 31 | \( 1 + 8.78T + 31T^{2} \) |
| 37 | \( 1 - 5.04T + 37T^{2} \) |
| 41 | \( 1 - 4.15T + 41T^{2} \) |
| 43 | \( 1 - 4.10T + 43T^{2} \) |
| 47 | \( 1 - 8.03T + 47T^{2} \) |
| 53 | \( 1 - 1.23T + 53T^{2} \) |
| 59 | \( 1 + 11.0T + 59T^{2} \) |
| 61 | \( 1 + 11.8T + 61T^{2} \) |
| 67 | \( 1 + 13.4T + 67T^{2} \) |
| 71 | \( 1 - 5.58T + 71T^{2} \) |
| 73 | \( 1 + 3.39T + 73T^{2} \) |
| 79 | \( 1 + 1.34T + 79T^{2} \) |
| 83 | \( 1 - 3.79T + 83T^{2} \) |
| 89 | \( 1 - 16.8T + 89T^{2} \) |
| 97 | \( 1 + 8.03T + 97T^{2} \) |
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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
−9.284243566003844663256357361717, −7.890825132507685695989759784196, −7.58706083730349298277597484439, −7.24746910498053463584639465590, −5.81375260253452150596053328473, −5.13049731808050976830498033239, −4.26297397015875186113455821645, −2.90170949905393631870208395875, −1.74124927566829554642632483060, −0.53286283152280864738514621979,
0.53286283152280864738514621979, 1.74124927566829554642632483060, 2.90170949905393631870208395875, 4.26297397015875186113455821645, 5.13049731808050976830498033239, 5.81375260253452150596053328473, 7.24746910498053463584639465590, 7.58706083730349298277597484439, 7.890825132507685695989759784196, 9.284243566003844663256357361717