L(s) = 1 | + 2-s − 1.95·3-s + 4-s − 1.56·5-s − 1.95·6-s + 1.47·7-s + 8-s + 0.826·9-s − 1.56·10-s + 11-s − 1.95·12-s − 1.42·13-s + 1.47·14-s + 3.05·15-s + 16-s − 5.43·17-s + 0.826·18-s + 3.19·19-s − 1.56·20-s − 2.88·21-s + 22-s + 5.12·23-s − 1.95·24-s − 2.55·25-s − 1.42·26-s + 4.25·27-s + 1.47·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1.12·3-s + 0.5·4-s − 0.698·5-s − 0.798·6-s + 0.556·7-s + 0.353·8-s + 0.275·9-s − 0.494·10-s + 0.301·11-s − 0.564·12-s − 0.395·13-s + 0.393·14-s + 0.789·15-s + 0.250·16-s − 1.31·17-s + 0.194·18-s + 0.732·19-s − 0.349·20-s − 0.628·21-s + 0.213·22-s + 1.06·23-s − 0.399·24-s − 0.511·25-s − 0.279·26-s + 0.818·27-s + 0.278·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 197 | \( 1 + T \) |
good | 3 | \( 1 + 1.95T + 3T^{2} \) |
| 5 | \( 1 + 1.56T + 5T^{2} \) |
| 7 | \( 1 - 1.47T + 7T^{2} \) |
| 13 | \( 1 + 1.42T + 13T^{2} \) |
| 17 | \( 1 + 5.43T + 17T^{2} \) |
| 19 | \( 1 - 3.19T + 19T^{2} \) |
| 23 | \( 1 - 5.12T + 23T^{2} \) |
| 29 | \( 1 - 2.49T + 29T^{2} \) |
| 31 | \( 1 - 0.687T + 31T^{2} \) |
| 37 | \( 1 + 3.11T + 37T^{2} \) |
| 41 | \( 1 + 9.26T + 41T^{2} \) |
| 43 | \( 1 - 1.12T + 43T^{2} \) |
| 47 | \( 1 - 2.17T + 47T^{2} \) |
| 53 | \( 1 - 8.13T + 53T^{2} \) |
| 59 | \( 1 + 9.33T + 59T^{2} \) |
| 61 | \( 1 - 3.38T + 61T^{2} \) |
| 67 | \( 1 + 3.81T + 67T^{2} \) |
| 71 | \( 1 - 2.58T + 71T^{2} \) |
| 73 | \( 1 + 6.76T + 73T^{2} \) |
| 79 | \( 1 - 3.30T + 79T^{2} \) |
| 83 | \( 1 - 7.08T + 83T^{2} \) |
| 89 | \( 1 - 16.5T + 89T^{2} \) |
| 97 | \( 1 + 8.76T + 97T^{2} \) |
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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
−7.85225238410868214634880706055, −6.96618494294140578861162807993, −6.59263885913515374003389248991, −5.62015423806890725667538064901, −4.98338801372944080749445186093, −4.49223059390693166243765993686, −3.57821686324760066039429395581, −2.56310785487674215424947584918, −1.31616983361749065643852744127, 0,
1.31616983361749065643852744127, 2.56310785487674215424947584918, 3.57821686324760066039429395581, 4.49223059390693166243765993686, 4.98338801372944080749445186093, 5.62015423806890725667538064901, 6.59263885913515374003389248991, 6.96618494294140578861162807993, 7.85225238410868214634880706055