L(s) = 1 | − 2.47·2-s + 3-s + 4.14·4-s − 2.47·6-s − 2.83·7-s − 5.30·8-s + 9-s + 6.09·11-s + 4.14·12-s + 3.00·13-s + 7.02·14-s + 4.86·16-s + 2.74·17-s − 2.47·18-s + 8.03·19-s − 2.83·21-s − 15.1·22-s + 7.56·23-s − 5.30·24-s − 7.43·26-s + 27-s − 11.7·28-s − 29-s − 5.32·31-s − 1.44·32-s + 6.09·33-s − 6.79·34-s + ⋯ |
L(s) = 1 | − 1.75·2-s + 0.577·3-s + 2.07·4-s − 1.01·6-s − 1.07·7-s − 1.87·8-s + 0.333·9-s + 1.83·11-s + 1.19·12-s + 0.832·13-s + 1.87·14-s + 1.21·16-s + 0.664·17-s − 0.584·18-s + 1.84·19-s − 0.618·21-s − 3.22·22-s + 1.57·23-s − 1.08·24-s − 1.45·26-s + 0.192·27-s − 2.21·28-s − 0.185·29-s − 0.955·31-s − 0.255·32-s + 1.06·33-s − 1.16·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.082812135\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.082812135\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 29 | \( 1 + T \) |
good | 2 | \( 1 + 2.47T + 2T^{2} \) |
| 7 | \( 1 + 2.83T + 7T^{2} \) |
| 11 | \( 1 - 6.09T + 11T^{2} \) |
| 13 | \( 1 - 3.00T + 13T^{2} \) |
| 17 | \( 1 - 2.74T + 17T^{2} \) |
| 19 | \( 1 - 8.03T + 19T^{2} \) |
| 23 | \( 1 - 7.56T + 23T^{2} \) |
| 31 | \( 1 + 5.32T + 31T^{2} \) |
| 37 | \( 1 + 6.28T + 37T^{2} \) |
| 41 | \( 1 + 9.04T + 41T^{2} \) |
| 43 | \( 1 + 9.06T + 43T^{2} \) |
| 47 | \( 1 - 0.0839T + 47T^{2} \) |
| 53 | \( 1 + 3.82T + 53T^{2} \) |
| 59 | \( 1 - 11.9T + 59T^{2} \) |
| 61 | \( 1 + 0.275T + 61T^{2} \) |
| 67 | \( 1 + 7.36T + 67T^{2} \) |
| 71 | \( 1 - 15.2T + 71T^{2} \) |
| 73 | \( 1 - 8.77T + 73T^{2} \) |
| 79 | \( 1 - 1.64T + 79T^{2} \) |
| 83 | \( 1 - 1.38T + 83T^{2} \) |
| 89 | \( 1 + 1.05T + 89T^{2} \) |
| 97 | \( 1 - 9.18T + 97T^{2} \) |
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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
−9.046516776942070549927265557045, −8.649888309574938671710872052215, −7.64722829957188565052078113897, −6.82302620325241651382512794166, −6.59910361158590620883203235904, −5.31683611640129005011865889786, −3.50368996007566118267747892512, −3.28802163077325270752652256117, −1.68332697349680870697875235883, −0.939099081887040149236553067373,
0.939099081887040149236553067373, 1.68332697349680870697875235883, 3.28802163077325270752652256117, 3.50368996007566118267747892512, 5.31683611640129005011865889786, 6.59910361158590620883203235904, 6.82302620325241651382512794166, 7.64722829957188565052078113897, 8.649888309574938671710872052215, 9.046516776942070549927265557045