L(s) = 1 | + 0.571·3-s − 2.42·5-s − 3.89·7-s − 2.67·9-s + 1.06·11-s + 5.87·13-s − 1.38·15-s + 4.69·17-s − 4.56·19-s − 2.22·21-s + 6.88·23-s + 0.867·25-s − 3.24·27-s − 4.73·29-s + 5.09·31-s + 0.606·33-s + 9.43·35-s + 7.53·37-s + 3.35·39-s − 4.59·41-s + 7.94·43-s + 6.47·45-s + 10.0·47-s + 8.17·49-s + 2.68·51-s + 7.90·53-s − 2.57·55-s + ⋯ |
L(s) = 1 | + 0.330·3-s − 1.08·5-s − 1.47·7-s − 0.890·9-s + 0.319·11-s + 1.62·13-s − 0.357·15-s + 1.13·17-s − 1.04·19-s − 0.486·21-s + 1.43·23-s + 0.173·25-s − 0.624·27-s − 0.878·29-s + 0.914·31-s + 0.105·33-s + 1.59·35-s + 1.23·37-s + 0.537·39-s − 0.718·41-s + 1.21·43-s + 0.965·45-s + 1.46·47-s + 1.16·49-s + 0.375·51-s + 1.08·53-s − 0.346·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.170756110\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.170756110\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 167 | \( 1 + T \) |
good | 3 | \( 1 - 0.571T + 3T^{2} \) |
| 5 | \( 1 + 2.42T + 5T^{2} \) |
| 7 | \( 1 + 3.89T + 7T^{2} \) |
| 11 | \( 1 - 1.06T + 11T^{2} \) |
| 13 | \( 1 - 5.87T + 13T^{2} \) |
| 17 | \( 1 - 4.69T + 17T^{2} \) |
| 19 | \( 1 + 4.56T + 19T^{2} \) |
| 23 | \( 1 - 6.88T + 23T^{2} \) |
| 29 | \( 1 + 4.73T + 29T^{2} \) |
| 31 | \( 1 - 5.09T + 31T^{2} \) |
| 37 | \( 1 - 7.53T + 37T^{2} \) |
| 41 | \( 1 + 4.59T + 41T^{2} \) |
| 43 | \( 1 - 7.94T + 43T^{2} \) |
| 47 | \( 1 - 10.0T + 47T^{2} \) |
| 53 | \( 1 - 7.90T + 53T^{2} \) |
| 59 | \( 1 + 7.93T + 59T^{2} \) |
| 61 | \( 1 - 12.9T + 61T^{2} \) |
| 67 | \( 1 + 6.91T + 67T^{2} \) |
| 71 | \( 1 + 5.67T + 71T^{2} \) |
| 73 | \( 1 - 14.1T + 73T^{2} \) |
| 79 | \( 1 + 5.05T + 79T^{2} \) |
| 83 | \( 1 + 10.2T + 83T^{2} \) |
| 89 | \( 1 - 12.7T + 89T^{2} \) |
| 97 | \( 1 + 10.4T + 97T^{2} \) |
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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
−9.400725425692048738233082706666, −8.797462330580167734176756945206, −8.124450220077516943744114970910, −7.21243266690833231437888507554, −6.28413122699739751769307866081, −5.68257123124367428124744754022, −4.09010009179829999778041986982, −3.53230383712891784405417157931, −2.76652621113286438428306661756, −0.76679874117256961077544409395,
0.76679874117256961077544409395, 2.76652621113286438428306661756, 3.53230383712891784405417157931, 4.09010009179829999778041986982, 5.68257123124367428124744754022, 6.28413122699739751769307866081, 7.21243266690833231437888507554, 8.124450220077516943744114970910, 8.797462330580167734176756945206, 9.400725425692048738233082706666